DIFFERENTIAL GEOMETRY BASED MULTISCALE MODELING OF SOLVATION
By
Zhan Chen

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Applied Mathematics
2011

ABSTRACT
DIFFERENTIAL GEOMETRY BASED MULTISCALE MODELING OF
SOLVATION
By
Zhan Chen
Solvation is an elementary process in nature and is of paramount importance to many sophisticated chemical, biological and biomolecular processes. The understanding of solvation is an
essential prerequisite for the quantitative description and analysis of biomolecular systems.
Implicit solvent models, particularly those based on the Poisson-Boltzmann (PB) equation
for electrostatic analysis, are established approaches for solvation analysis. However, ad hoc
solvent-solute interfaces are commonly used in the implicit solvent theory and have some
severe limitations.
We have introduced differential geometry based solvation models which allow the solventsolute interface to be determined by the variation of a total free energy functional. Our
models extend the scaled particle theory (SPT) of nonpolar solvation models with a solventsolute interaction potential. The nonpolar solvation model is completed with a PB theory
based polar solvation model. In our Eulerian formation, the differential geometry theory of
hypersurface is utilized to define and construct smooth interfaces with good stability and
differentiability, for use in characterizing the solvent-solute boundaries and in generating
continuous dielectric functions across the computational domain. Some techniques from the
geometric measure theory are employed to rigorously convert a Lagrangian formulation of
the surface energy into an Eulerian formulation, so as to bring all energy terms on an equal
footing. In our Lagrangian formulation, the differential geometry theory of surfaces is used to
provide a natural description of solvent-solute interfaces. By optimizing the total free energy

functional, we derive a coupling of the generalized Poisson-Boltzmann equation (GPBE) and
the generalized geometric flow equation (GGFE or also called Laplace-Beltrami equation)
for the electrostatic potential and the construction of realistic solvent-solute boundaries,
respectively. The coupled partial differential equations (PDEs) are solved with iterative
procedures to reach a steady state, which delivers the desired solvent-solute interface and
electrostatic potential for many problems of interest. These quantities are utilized to evaluate
the solvation free energies, protein-protein binding affinities, etc.
The above proposed approaches have been extensively validated.Extensive numerical experiments have been designed to validate the present theoretical models, to test the computational methods, and to optimize the numerical algorithms. Solvation analysis of both
small compounds and proteins are carried out to further demonstrate the accuracy, stability,
efficiency and robustness of the present new models and numerical approaches. Comparison
is given to both experimental and theoretical results in the literature.
Moreover, to account for the charge rearrangement during the solvation process, we also
propose a differential geometry based multiscale solvation model which makes use of electron
densities computed directly from a quantum mechanical approach. We construct a new total
energy functional, which consists of not only polar and nonpolar solvation contributions, but
also the electronic kinetic and potential energies. We show that the quantum formulation
of our solvation model improves the prediction of our earlier models, and outperforms some
explicit solvation analysis.

ACKNOWLEDGMENT

First of all, I would like to express my deepest and sincerest gratitude to my dissertation
advisor, Dr. Guowei Wei for his invaluable supports, guidance and personal care. It is
beyond my words of appreciation that he is a far-sighted person with a broad interdisciplinary
perspective, and that he has a unique creative insight which often sheds light on undiscovered
potentialities of a subject. More than being a polymath, he cares about his graduate students
so much that I have not seen any other advisor like him. Throughout these years, he has
been cultivating my passion in science and academic aspirations. He has enormous patience
on me, even during the most challenging times in my Ph.D. study. His spirit of pursuing
academic dreams and perfection has a life-long influence on me. It has been an enormously
enjoyable experience working with him.
My gratitude also goes to my defense committee members, Dr. Changyi Wang, Dr. Leslie
Kuhn, Dr. Chichia Chiu, and Dr. Moxun Tang for their expertise and precious time. I also
would like to thank Dr. Zhengfang Zhou for his generous supports and unceasing cares for
my wife and me throughout our Ph.D. program, and Dr. Keith Promislow for his kindness
and the inspiring classroom discussions, and Dr. Nathan Baker for his supports and intense
interests in our work of sovlation.
Moreover, I would like to thank Dr. Shan Zhao for his continuing helps and encouragement in my research. Dr. Zhao is the one who taught me how to write FORTRAN codes
after I joined this group. My thanks also go to the rest of current and former members in Dr.
Wei’s group: Dr. Yongcheng Zhou, Dr. Yuhui Sun, Dr. Sining Yu, Dr Weihua, Geng, Dr.
Duan Chen, Dr. Siyang Yang, Ms. Qiong Zheng, Mr. Langhua Hu, Ms. Jin Kyoung Park,
iv

Ms. Yajie Yu, Ms. Weijuan Zhou, Mr. Wangheng Liu, Mr. Manfeng Hu, Ms. Shuailing
Wang, Mr. Qi Zhao, Mr. Kelin Xia, Mr. Huibing Zhu, for their valuable discussions and
helps in my research and daily life. Our group is like a big loving family. I will always
treasure our friendships. I am also grateful to Ms. Barbara Miller, Graduate Secretary in
the Department of Mathematics, for her generous helps during my graduate study, and to
Mr. Nick Boros for proofreading a part of my dissertation.
Finally, from my very deep heart I thank Yuting, my wife and loved one, for her constant
love, understanding and supports throughout my Ph.D. study. She is always patient to me
and always has trusts in me. To her, words can not express my gratitude.

v

To my family: Baokun, Ziying, Yan, Xiang and Yuting

vi

TABLE OF CONTENTS

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of Figures

x

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1 Introduction
1.1 Introduction to solvation models . . . . . . . . . . . . . . . . . .
1.1.1 Biological background . . . . . . . . . . . . . . . . . . .
1.1.2 Polar solvation models . . . . . . . . . . . . . . . . . . .
1.1.3 Poisson-Boltzmann theory . . . . . . . . . . . . . . . . .
1.1.4 Nonpolar solvation models . . . . . . . . . . . . . . . . .
1.2 Molecular interface definitions . . . . . . . . . . . . . . . . . . .
1.3 Quantum mechanical continuum models . . . . . . . . . . . . .
1.4 Limitations of current models . . . . . . . . . . . . . . . . . . .
1.5 Mathematical issues and numerical challenges . . . . . . . . . .
1.5.1 Geometry, PDE and interface . . . . . . . . . . . . . . .
1.5.2 Geometric flow equation . . . . . . . . . . . . . . . . . .
1.5.3 Highly accurate and efficient solver for interface problems
1.5.4 Self-consistent iterative methods . . . . . . . . . . . . . .
1.6 The rest of this thesis . . . . . . . . . . . . . . . . . . . . . . . .

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2 Eulerian formulation
2.1 Theory and model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Solute-solvent boundary . . . . . . . . . . . . . . . . . . . . .
2.1.2 Total free energy functional . . . . . . . . . . . . . . . . . . .
2.1.2.1
Polar free energy functional . . . . . . . . . . . . . .
2.1.2.2 Non-polar free energy functional . . . . . . . . . . .
2.1.3 Governing equations . . . . . . . . . . . . . . . . . . . . . . .
2.2 Methods and algorithms . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Discretization schemes of the governing equations . . . . . . .
2.2.1.1 The generalized Poisson-Boltzmann equation . . . .
2.2.2 Acceleration procedures . . . . . . . . . . . . . . . . . . . . .
2.2.2.1 Precondition of the PB solver . . . . . . . . . . . . .
2.2.2.2 Initial guess of the generalized PB solution . . . . . .
2.2.2.3 Convergence criteria in the generalized PB solver . .
2.2.3 Dynamical coupling of the generalized Poisson Boltzmann and
etry flow equations . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3.1 Approach I . . . . . . . . . . . . . . . . . . . . . . .
vii

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geom. . . .
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xii
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2.4

2.5

2.2.3.2 Approach II . . . . . . . . . . . . . . . . . . .
2.2.4 Evaluation of the solvation free energy . . . . . . . . .
Numerical test and validation . . . . . . . . . . . . . . . . . .
2.3.1 The behavior of the coarea formula . . . . . . . . . . .
2.3.2 Accuracy of the generalized PB solver . . . . . . . . . .
2.3.3 Convergence of boundary profile and dielectric function
2.3.4 Consistency of iteration procedures . . . . . . . . . . .
2.3.5 Efficiency of the accelerated iteration procedure . . . .
2.3.6 Impact of potentials in the geometric flow equation . .
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Set of 17 test molecules . . . . . . . . . . . . . . . . . .
2.4.2 Solvation free energy of proteins . . . . . . . . . . . . .
2.4.3 Twenty two proteins . . . . . . . . . . . . . . . . . . .
Chapter conclusions . . . . . . . . . . . . . . . . . . . . . . . .

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3 Quantum formulation
3.1 Theory and model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Charge density based polar free energy functional . . . . . . . . . .
3.1.2 Quantum mechanical energy functional . . . . . . . . . . . . . . . .
3.1.2.1 Kinetic energy . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2.2 Potential energy . . . . . . . . . . . . . . . . . . . . . . .
3.1.3 Total free energy functional . . . . . . . . . . . . . . . . . . . . . .
3.1.4 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Numerical methods and algorithms . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Solution of the generalized Poisson-Boltzmann equation . . . . . . .
3.2.2 Solution of the generalized Kohn-Sham equation . . . . . . . . . . .
3.2.3 Evaluation of the solvation free energy . . . . . . . . . . . . . . . .
3.2.4 Dynamical coupling of involved PDE equations . . . . . . . . . . .
3.3 Numerical test and validation . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Validation of the cancellation of self-interaction energy . . . . . . .
3.3.2 Validation of data translation and unit conversion . . . . . . . . . .
3.3.3 Accuracy of solvation free energies computed by the present model .
3.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Solvation free energies of 24 small molecules . . . . . . . . . . . . .
3.4.2 Solvation free energies of 16 molecules . . . . . . . . . . . . . . . .
3.4.3 Solvation free energies of 3 larger molecules . . . . . . . . . . . . .
3.5 Chapter conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Lagrangian formulation
4.1 Theory and model . . . . . . . . . . . . . . .
4.1.1 Solvation free energy functionals . . . .
4.1.1.1 Polar solvation functional . .
4.1.1.2 Total free energy functional of
4.1.2 Surface variation . . . . . . . . . . . .
viii

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solvation
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4.2

4.3

4.4

4.5

4.1.3 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . .
Methods and algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Interconversion between the Lagrangian and Eulerian representations
4.2.1.1 Embedding the Lagrangian dynamics into the Eulerian representation . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1.2 Transform from the Lagrangian representation to the Eulerian representation . . . . . . . . . . . . . . . . . . . . . . .
4.2.1.3 Transform from the Eulerian representation to the Lagrangian
representation . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1.4 Numerical surface integral and volume integral in the Eulerian representation . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Dynamic coupling of the Poisson-Boltzmann and geometric flow equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Validation of the interface extraction scheme . . . . . . . . . . . . . .
4.3.2 Effect of interaction potentials . . . . . . . . . . . . . . . . . . . . . .
4.3.2.1 Surfaces of a diatom system . . . . . . . . . . . . . . . . . .
4.3.2.2 Surfaces of a four-atom system . . . . . . . . . . . . . . . .
4.3.2.3 Surfaces and electrostatic potentials of a protein . . . . . . .
4.3.3 Isosurface function value and minimal surfaces . . . . . . . . . . . . .
4.3.4 Convergence of surface area, volume and energy . . . . . . . . . . . .
Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Free energy calculations . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1.1 Solvation energies of 17 compounds . . . . . . . . . . . . . .
4.4.1.2 A set of 23 proteins . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Salt effect on protein-protein binding energies . . . . . . . . . . . . .
Chapter conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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167

5 Thesis achievements and future work
170
5.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
5.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
A Solution of the generalized Laplace-Beltrami equation and the ADI scheme178
B PB equation in different forms

185

C Differential geometry theory preliminary

188

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

192

ix

LIST OF TABLES

2.1

Areas computed from the coarea formula for bounded open sets . . . . . . .

48

2.2

Errors and convergence orders for the generalized PB solver (ǫ1 = 1)

. . . .

51

2.3

Comparison between two iteration approaches . . . . . . . . . . . . . . . . .

56

2.4

Effect of relaxation factor α on final results . . . . . . . . . . . . . . . . . . .

57

2.5

Effect of the number of intermittency in Approach II . . . . . . . . . . . . .

58

2.6

CPU time analysis from original schemes . . . . . . . . . . . . . . . . . . . .

59

2.7

Speedup from adjustment of initial guess and preconditioner in PB solver . .

59

2.8

Influence of convergence criteria on electrostatic solvation free energy and
computational time for the diatomic system . . . . . . . . . . . . . . . . . .

60

Comparison of CPU time (s) in the iteration procedures with and without
accelerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

2.10 Effects of potentials on the solvent-solute boundary . . . . . . . . . . . . . .

63

2.11 Comparison of free energies (kcal/mol) for 17 compounds . . . . . . . . . . .

64

2.12 Comparison of electrostatic solvation free energies (kcal/mol) obtained from
the MFCC-CPCM, the present model (OSM) and MIBPB. . . . . . . . . . .

68

2.13 Electrostatic solvation free energies for 22 proteins . . . . . . . . . . . . . . .

71

2.9

3.1

Comparison of total electrostatic energy (kcal/mol) and electrostatic solvation
energy (kcal/mol) obtained with the partial charge approach. . . . . . . . . . 104

3.2

Comparison of total electrostatic energy (kcal/mol) and electrostatic solvation
energy (kcal/mol) obtained with the direct use of charge density. . . . . . . . 105
x

3.3

Comparison of solvation energy components between present results and those
of Wang et al [238] for three small molecules. . . . . . . . . . . . . . . . . . . 106

3.4

Solvation free energy (kcal/mol) and its decomposition. . . . . . . . . . . . . 107

3.5

Comparison of solvation free energies (kcal/mol) obtained from the present
model and experimental data for 24 small molecules. . . . . . . . . . . . . . 108

3.6

Solvation free energy (kcal/mol) decomposition for a set of 21 molecules. . . 112

3.7

Comparison of free energies (kcal/mol) for 16 compounds. . . . . . . . . . . 114

3.8

Free energies (kcal/mol) for 16 compounds using structures from Pubchem
data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

3.9

Solvation free energies (kcal/mol) of 3 large molecules and corresponding CPU
time (hour). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.1

Comparison of surface areas (˚2 ), volumes (˚3 ) and energies (kcal/mol) for
A
A
two small systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

4.2

˚
Electrostatic solvation free energies (kcal/mol), surface areas (A2 ) and vol3 ) of protein 451c with different solvent-solute interactions. . . . . . 151
umes (˚
A

4.3

Surface areas (˚2 ) for different surface definitions . . . . . . . . . . . . . . . 153
A

4.4

RMS error with different nonpolar potentials. . . . . . . . . . . . . . . . . . 158

4.5

Predicted and experimental total solvation free energies for 17 small compounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

4.6

Comparison of electrostatic solvation free energies of 23 proteins. . . . . . . . 162

4.7

Comparison of binding affinities of two proteins complexes from current simulations and those from Bertonati et al’s paper. . . . . . . . . . . . . . . . . 166

xi

LIST OF FIGURES

1.1

A solvation free energy cycle adapted from Levy et al. [130]. The total
solvation energy (1) is decomposed into several steps: “charging” the solute
in solvent (6) and vacuum (2), including attractive dispersive solute-solvent
interactions in solvent (5) and vacuum (3), and cavity formation associated
with repulsive solute-solvent interactions (4). The energy associated with Step
(7) is generally termed a “nonpolar solvation energy” while the difference in
energies associated with Steps (1) and (7) is generally considered as “polar
solvation energy”. For interpretation of the references to color in this and all
other figures, the reader is referred to the electronic version of this dissertation.

4

2.1

The cross line of S and (1 − S) of a diatomic system described in Section 2.3.3 29

2.2

The cross line profile of ǫ(S) of a diatomic system described in Section 2.3.3.
Here, we have set ǫs = 80 and ǫm = 1. . . . . . . . . . . . . . . . . . . . . .

36

The evolutionary profiles of the S function at cross section (x, y, 0.05) in a
diatomic system plotted from four intermediate states. . . . . . . . . . . . .

53

2.3

2.4

The time evolution histories of the electrostatic solvation free energy, F (Volume)
and J(Area), where F (Volume) = volume/5−180 and J(Area) = (surface area)/5−
200. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.5

Correlation between experimental data and the present optimized surface
model (OSM)(also results from Nicholls’) in electrostatic solvation free energies of 17 compounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

Correlation between MFCC-CPCM [152] and the present optimized surface
model (OSM) in electrostatic solvation free energies of 8 proteins. . . . . . .

69

Correlation between MIBPB-III and the present model (OSM) in electrostatic
solvation free energies of 22 proteins . . . . . . . . . . . . . . . . . . . . . . .

70

Differences between electrostatic solvation free energies obtained from the
MIBPB and the present model with original radii (Radii0) or enlarged radii
(Radii1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

2.6

2.7

2.8

xii

2.9

Surface potential display of one protein (PDBID: 1frd) at different isosurfaces

72

3.1

Flowchart of the numerical solution of the coupled PDEs. . . . . . . . . . . .

99

3.2

Correlation between experimental data and the present optimized surface
model with quantum correction (OSMQ) in solvation free energies of 24 small
molecules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

3.3

Illustration of surface electrostatic potentials of four small compounds at
their corresponding isosurfaces S = 0.50. (a) Glycerol triacetate; (b) 1,1diethoxyethane; (c) Bis-2-chloroethyl ether; (d) Dimethoxymethane. . . . . 111

3.4

Correlation between experimental data [160] and the present optimized surface model with quantum mechanics (OSMQ) in solvation free energies of 16
compounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

3.5

Illustration of surface electrostatic potentials of three compounds at their
corresponding isosurfaces S = 0.50. (a) Phorbol; (b) Phorbol12,13-dibutyrate;
(c) Staurosporine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.1

˚
Illustration of surface morphology of a diatom system with radii 1.87A and
coordinates (x, y, z) = (−2.2, 0, 0), and (2.2, 0, 0) under different solute-solvent
interactions. Top Left: The MMS (no potential); Top Right: The surface obrep,LJ
tained under a repulsive potential (Vi
); Bottom Left: The surface obtained under a full L-J potential (V LJ ); Bottom Right: The surface obtained
under a full L-J potential and an electrostatic potential. It can be seen that
the repulsive potential produces a “fat” surface, while an attractive potential
or an electrostatic potential leads to a “slim” surface. . . . . . . . . . . . . . 147

4.2

Illustration of surface morphology of a four-atom system with radii 1.87˚ and
A
coordinates (x, y, z) = (−3.40, 0, 0), (3.40, 0, 0), (0, −2.94, 0) and (0, 2.94, 0)
under different solute-solvent interactions. Top Right: The surface obtained
rep,LJ
under a repulsive potential (Vi
); Bottom Left: The surface obtained
under the full L-J potential (V LJ ); Bottom Right: The surface obtained
under the full L-J potential and an electrostatic potential. Again, it can be
seen that the repulsive potential producess a “fat” surface; while an attractive
potential or an electrostatic potential leads to a “slim” surface. . . . . . . . 149
xiii

4.3

The projection of electrostatic surface potentials of protein 451c onto different
surfaces obtained under various solvent-solute interactions. Top Left: Attractive surface; Top Right: The MMS; Bottom Left: Repulsive surface; Bottom
Right: Polar surface. It is noted that the repulsive surface is a “fat” surface
comparing to the MMS; while an attractive surface or a polar surface is a
“slim” surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

4.4

˚
Difference of surface areas ( A2 ) between MMS and various resulting surfaces
generated under different constant pressure effects. . . . . . . . . . . . . . . 155

4.5

Decreasing of surface area (×102 ˚2 ), J(volume)=(volume-2) (×102 ˚3 ) and
A
A
total solvation free energy (kcal/mol) in diethyl propanedioate as the number
of iterations increases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

4.6

Correlation of solvation free energy between previous optimized smooth surface (OSS) model and the present optimized molecular surface (OMS) model
in the set of 17 compounds listed in Table 4.5. . . . . . . . . . . . . . . . . 159

4.7

Correlation of electrostatic solvation free energy between the present optimized molecular surface (OMS) model, and previous models, such as the optimized smooth surface (OSS), the MIBPB-III and the MMS for 23 proteins
listed in Table 4.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

4.8

Difference of electrostatic solvation free energies between the OMS model and
previous OSS and MMS models for 23 proteins listed in Table 4.6. . . . . . . 163

4.9

Protein-protein complexes. Left: Protein complex 1emv; Right: Protein complex 1beb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

4.10 The salt dependence of the binding affinities of two protein complexes. Left:
Protein lemv; Right: Protein 1beb. Here OMS data are computed by our optimized molecular surface (OMS) model. NLPB data are taken from Bertonati
et al’s paper [26]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

xiv

Chapter 1
Introduction

1.1
1.1.1

Introduction to solvation models
Biological background

Almost all important biological processes in nature, including signal transduction, DNA
recognition, transcription, post-translational modification, translation, protein folding and
protein ligand binding, naturally occur in water, which comprises 65-90% of cellular mass.
The understanding of solvation is an elementary prerequisite for the quantitative description
and analysis of the above-mentioned processes. Solvation involves the energetics of interactions between solute molecules and solvent molecules or ions in the aqueous environment.
Solute-solvent interactions are typically described by solvation energies (or closely related
quantities): the free energy of transferring the solute from a vacuum to the solvent environment of interest (e.g.,water at a certain ionic strength), as shown in more detail in Figure1.1.
Solvation free energy is a physical quantity that can be measured experimentally.
Although millions of organic compounds are known now, only several thousands of com1

pounds have experimental data being reported for the solvation free energy. It is mainly due
to experimental difficulties associated with the precise measurement, particularly for those
compounds with low solubility and/or low volatility [182, 174]. Because of low solubility
and/or low volatility, accurate and time-consuming measurement is required with highly
sensitive instruments. Unfortunately, many important organic compounds belong to this
category. Moreover, attentions need to paid on the chemical stability of solute under investigation. Therefore, the experimental study of solvation free energy still remains expensive,
laborious and is sometimes inaccurate.
Computational approaches provide an alternative method to obtain the solvation free
energy. Solvation free energies can be calculated by a variety of computational methods,
ranging from very time-consuming quantum mechanical approaches [111, 183, 148, 118] to
simple phenomenological modifications of Coulomb’s law. Solvation models can be roughly
divided into two main classes [186, 239, 203, 200]: explicit solvent models that describe the
solvent in molecular or atomic detail [179], and implicit solvent models that generally replace
the explicit solvent with a dielectric continuum [8, 10, 65, 109, 186, 117]. Explicit solvent
models provide the detailed information on molecular constitutions, and generally require
extensive sampling to extract meaningful thermodynamic, statistical or kinetic properties of
interest. Whereas, implicit solvent models focus on the biomolecules of interest, and take a
mean field approximation for solvent properties. Because of their fewer degrees of freedom,
implicit solvent methods have become popular for many applications in molecular simulation
[7, 82, 9, 69].
To help the calculation of solvation energy, one can conceptually break up the solvation process as follows: #1 in this figure can be decomposed into two basic processes: a
2

“nonpolar” process of inserting the uncharged solute into solvent (#7) and a “polar” process of charging the solute in vacuum (#2) and solvent (#6). The free energy change in
#7 is called the nonpolar solvation energy. The difference of energies associated with #6
and #2 is called the “charging” or polar solvation energy and represents the solvent’s effect
on the solute charging process. The polar portion of solvation originates from electrostatic
interactions, which are ubiquitous for any system of charged or polar molecules, such as
biomolecules (proteins, nucleic acids, lipid bilayers, sugars, etc.) in their aqueous environment [240, 65, 69, 105, 200, 239, 203, 86, 204, 7, 82, 9]. The nonpolar portion describes
the remaining contributions, including the surface tension, mechanical work, and attractive
solvent-solute dispersion interactions.

1.1.2

Polar solvation models

Electrostatic interactions are ubiquitous in nature. For biomolecular systems in aqueous
environment, the analysis of molecular solvation and electrostatics is of great importance to
research in chemistry, biophysics, medicine and nano-technology. Implicit solvent models are
widely used in such an analysis which can be classified into two general types: quantitative
analysis and qualitative study. One of the primary quantitative application in computational biology and chemistry has been the calculation of thermodynamic properties. Implicit
solvent methods “pre-equilibrate” the solvent and mobile ions, thus effectively pre-compute
the solvent contribution for a system [186]. Such pre-equilibration is particularly evident in
MM/PBSA models [246, 216, 173, 220, 149], which combine implicit solvent approaches with
molecular mechanical models to evaluate binding free energies from an ensemble of biomolecular structures [16, 2, 133, 162, 145, 223, 161, 251, 162, 95, 150, 132, 131]. These methods
3

Figure 1.1: A solvation free energy cycle adapted from Levy et al. [130]. The total solvation
energy (1) is decomposed into several steps: “charging” the solute in solvent (6) and vacuum
(2), including attractive dispersive solute-solvent interactions in solvent (5) and vacuum
(3), and cavity formation associated with repulsive solute-solvent interactions (4). The
energy associated with Step (7) is generally termed a “nonpolar solvation energy” while the
difference in energies associated with Steps (1) and (7) is generally considered as “polar
solvation energy”. For interpretation of the references to color in this and all other figures,
the reader is referred to the electronic version of this dissertation.

4

have been employed to interpret experimental titration curves, analyze residue contributions in protein-protein and protein-ligand binding energetics, examine structural/functional
consequences of RNA nucleotide protonation, etc.

Another quantitative application of

implicit solvent models is the evaluation of biomolecular kinetics where implicit solvent
models are generally taken to compute solvation forces for molecular Langevin dynamics
[219, 180, 181, 142, 141], Brownian dynamics [147, 89, 76, 195], or continuum diffusion
[49, 50, 255, 210, 211] simulations. A major qualitative study of implicit solvent methods is
the visualization and qualitative analysis of electrostatic potentials on and around biomolecular surfaces [241, 177, 11, 7]. Visualization has become a standard procedure in the analysis
of biomolecular structures, including ligand-receptor binding, drug design, macromolecular
assembly, protein-nucleic acid complexes, protein-protein interactions, enzymatic mechanism
study, etc.

The polar solvation energy is generally associated with a difference in charging free energies in vacuum and solvent (see Figure 1.1 (#2) and (#6)). Polar solvation process and
electrostatic effect are described by a variety of implicit solvent models [240, 186, 239, 200,
65, 92, 120, 41, 193, 233, 198, 10, 9, 221]; however, the most widely-used ones are PoissonBoltzmann (PB) models [105, 69, 7, 125, 86, 200, 65, 117], generalized Born (GB) methods
[68, 15, 229, 167, 92, 263, 120, 226, 155, 41, 102] and polarizable continuum models (PCM)
[52, 227, 113, 218, 35, 14, 59]. Polarizable continuum models are proposed to model the
solvent either as polarizable dielectrics or as conductor-like media, and treat the solute compound by the quantum mechanical means [52, 227, 113, 218, 35, 14, 59]. These approaches
have often been used in reactively kinetics where quantum mechanical descriptions are desired. Generalized Born methods are relatively fast, but are not as accurate as the PB
5

methods [9, 166, 68, 166, 63, 230, 226]. They are often employed in high-throughput applications such as molecular dynamics [15, 229, 203, 82, 120, 68, 167, 63, 41]. PB methods can
be formally derived from more detailed theories [22, 159, 107] and provide a more accurate,
although somewhat slower, approach for evaluating polar solvation properties [63, 166, 15].
Moreover, unlike most generalized Born methods, PB models offer a global description for
the electrostatic properties, therefore making them uniquely suited to visualization and other
studies [138, 29, 66, 234, 89, 76, 64, 196, 211] where the electrostatic information is required
for both inside and outside a biomolecule.

1.1.3

Poisson-Boltzmann theory

Mathematically, the PB equation [105, 69, 7, 125, 86, 199, 65] is a nonlinear elliptic partial
differential equation (PDE) which is solved for the electrostatic potential. It is a continuum
model at equilibrium state, which dictates the solvent with a piecewise dielectric constant
and ionic charge density by the Boltzmann distribution. The PB equation can be derived by
the Gauss law and the Boltzmann distribution law [108]. Additioally, in the physical point of
view, the free energy of the system must be minimized at the equilibrium state. Therefore, a
total electrostatic free energy functional may be developed based on the PB theory, then the
PB equation can also be obtained by the variational principle [199]. The standard formula
of the PB equation is the following:
Nc
−∇ · (ǫ(r)∇φ) −

i=1

Qi n0 e−φQi /kB T =
i

Nm
j

qj δ(r − rj )

(1.1)

where ǫ is piecewise constant and depends on interface, being ǫs in solvent and ǫm in solute;
φ is electrostatic potential. Here qj is the partial charge on an atom located at xj , Qi is the
6

charge of ion species i, Nc is the number of ion species, kB is the Boltzmann constant, T is
the temperature, Nm is the total number of solute atoms, and n0 is the bulk concentration
i
of the ith ionic species. Note that the PB equation can appear in different forms according
to purposes as well as unit representation (detailed description can be found in Appendix B)

The PB theory is approximate and, as a result, has several well-known limitations which
can affect its accuracy [105, 159, 107, 69, 239, 198, 81, 56, 221, 222, 191, 45]. These limitations have been reviewed in the literature and will only be briefly summarized here. First,
most continuum models assume linear and local solvent response [239, 198, 81, 22]. However,
nonlinear solvent response (usually through dielectric saturation or electrostriction), can be
important in regions of strong electric field [239, 198, 81]. Biologically-relevant examples of
nonlinear solvent response have been found near highly charged ions, biomolecules, and other
interfaces. Nonlocal solvent response generally involves the finite non-zero size of water and
its unique hydrogen bonding with solute and other solvent molecules. Such nonlocal response
can be important in describing the orientation of water at biomolecular interfaces [38], differing solvation of cations and anions, and the solvation of asymmetric charge distributions.
The second major limitation is the mean-field treatment of ions in PB theory [107, 159, 105].
Mean field models assume that each ion experiences only the average influence of the other
ions in solution. Such averaging precludes detailed ion-ion interactions involving steric repulsion of ions (or their solvation shells) and Coulombic interaction of ions, including repulsion
and attractive pairing. The mean field assumption thereby eliminates correlations and fluctuations which can have important energetic and structural consequences for solutions of
divalent and multivalent ions surrounding highly-charged molecules such as nucleic acids
[56, 221, 222, 191, 45]. As suggested by the limitations above, PB models also neglect de7

tailed ion-solvent interactions which eliminate differences between ion species in solution and
thereby prevent effects analysis of specific ion species – which can be important in biophysical modeling. However, despite these limitations, PB methods are still very important for
biomolecular structural analysis, modeling, and simulation. Furthermore, these limitations
are currently being addressed through new implicit solvent models [5, 56, 159, 221, 175] and
hybrid treatments [232, 13, 128, 165, 156] which extend the applicability of the PB theory
while preserving some of its computational efficiency through pre-averaging solvent and ion
response.

1.1.4

Nonpolar solvation models

Poisson-Boltzmann methods provide polar solvation energies and therefore must be complemented by nonpolar solvation models to provide a complete view of solvent-solute interactions. As illustrated in Figure 1.1, nonpolar solvation is generally associated with the
insertion of the uncharged solute into solvent. There are many nonpolar solvation models
available. The most commonly used one is solvent-accessible surface area (SASA) models. They states that nonpolar solvent-solute interactions are proportional to the area of
the solvent-solute interface. It is worth to note that the proportional constant varies dramatically in the literature because the energies of other processes are also assumed to be
proportional to SASA [77]. Roughly speaking, SASA models are based on the scaled particle theory (SPT) [213, 178] which actually includes the energy of surface tension effect and
the mechanical work of immersing a particle into the solvent. Moreover, studies indicates
that nonpolar distribution should depend on the solvent-accessible volume and surface, with
a crossover to SASA when the size of solute is large [235]. Recent work by Levy, Gallic8

chio, and others [90, 92, 130, 91, 235] has demonstrated the importance of nonpolar solvent
models which include treatment of attractive solute-solvent dispersion terms (#5 in Figure
1.1) as well as models of solvent-solvent repulsive interactions (#4 in Figure 1.1), which are
described by both area and volume contributions [235]. Based on these considerations, in
the present work, we use the following model for nonpolar solvation free energies [235]

Gnp = γ · Area + p · Vol +

Ωs

ρs Uss dr,

(1.2)

where γ is the surface tension, ”Area” is the solvent-excluded area of the solute, p is the
hydrodynamic pressure, ”Vol” is the solvent-excluded volume of the solute, ρs is the solvent
density, Ωs denotes the solvent accessible region, and Uss is the solvent-solute van der Waals
(vdW) interaction potential. The first two terms in Eq. (1.2) are those from the SPT model
[213, 178]. The first term is the surface energy. It measures the disruption of intermolecular
and/or intramolecular bonds that occurs when a surface is created. The second term is the
mechanical work of creating the vacuum of a biomolecular size in the solvent. The third
term represents the attractive dispersion effects near the solvent-solute interface which has
been shown by Wagoner and Baker [235] to play a crucial role in accurate nonpolar solvation
analysis.In general, Uss can be obtained by the sum of the interaction of individual atoms
in Ωm with the solvent continuum in Ωs under the assumption that the nonpolar solutesolvent potential is pairwise: Uss =

vdW . This model of nonpolar solvation has
i Vi

been demonstrated to give good agreement with explicit solvent solvation forces on proteins
[235] and RNA hairpins [71]. Work by Levy and co-workers has demonstrated the good
performance of a similar model [90, 92, 130, 91].
In the present work, we further allow the solvent density ρs to be a function of position in
9

general. In particular, we split the solvent density ρs into the sum of atomic or ionic density
distribution functions ρs =

i ρs,i . The distribution of an individual solvent component can

be computed by integral equations or other approaches, such as Monte Carlo and generalized
Langevin equation [231, 88, 23]. This design of solvent density allows us to recover the
nonlinear and nonlocal effects of the solvent-solute interactions.

1.2

Molecular interface definitions

The separation of discrete and continuum domains in implicit solvent models requires an
interface to indicate the separation of solute atoms from the surrounding solvent. Naturally,
such an interface can be regarded as the surface or the profile of a molecule. The definition of
molecular profiles, or molecular graphics traces back to Corey and Pauling in 1950s [58], who
tried to depict the profiles of amino acids, peptides and proteins from X-ray crystallography.
In quantum chemistry, molecular graphics are often associated with the shapes of polynomial
functions that provide approximation to electron wavefunctions. In fact, since the electron
wavefunction changes its distribution under different environments, molecular profiles change
accordingly. Commonly used interface definitions in implicit solvent models include the
van der Waals surface, the solvent accessible surface [126], and the molecular surface (MS)
[185, 57]. In certain sense, these interface definitions determine the performance of implicit
solvent models because all of the physical properties of interest, including electrostatic free
energies, biomolecular surface areas, molecular cavitation volumes, solvation free energies,
and pKa values are very sensitive to these interface definitions [70, 72, 163, 217].
The use of PB model encounters some challenges in molecular dynamics regarding stability and accuracy. For example, the widely used molecular surface based PB model results
10

in forces which are unstable over time, lack analytical expression, are sensitive to grid discretization, and converge poorly [101]. Moreover, a discontinuous dielectric definition leads
to numerical instability regardless of the location of boundary [217]. Additionally, more
physically realistic surface definitions are desired because of the argument that the macroscopic physical properties must vary continuously. To overcome these difficulties, overlapping
atom-centered Gaussian or polynomial functions have been proposed to define the solute surface, with smooth transitions between low and high dielectric values [101, 112]. Although
continuous dielectric functions give rise to an improvement of stability and computational
efficiency, some recent work demonstrates that most of them are physically incorrect, which
will be discussed more in Section 1.4.

The recent development of a new class of molecular interfaces that incorporate the fundamental laws of physics starts with the construction of partial differential equation (PDE)
based molecular surface by Wei el al. in 2005 [245]. This approach distinguishes itself
from many other PDE based surface smoothing methods [249, 256] by utilizing only atomic
information, i.e., atomic coordinates and radii, instead of an existing surface. The atomic information is embedded in the Eulerian formulation and a family of hypersurfaces are evolved
in time under the PDE operator, which is designed to control the curvature and surface
tension. The generalized molecular surface is subsequently extracted from the final hypersurface by a level-set type of approach [245]. This PDE based surface construction procedure
generates well defined molecular surfaces for both small molecules and large proteins [245].
To our knowledge, geometric PDE based approach was the first of its kind for molecular surface construction. A further progress in the development of a “physical interface” was the
introduction of the minimal molecular surface (MMS) that minimizes a surface free energy
11

functional by the variational principle and leads to the mean curvature flow in 2006 [18, 19].
To our knowledge, MMSs were the first set of biomolecular surfaces that had ever been constructed by means of variational principles. The construction of the MMS was driven by the
desire to understand the true physical boundary of a biomolecule in solvent. As a physical
concept, the solvent-solute interface should be in general determined by the minimization of
the free energy of a macromolecule in the aquatic environment. The MMS is constructed by
using essentially the same procedure as that developed in the first PDE based surface generation method [245]. Another desirable property of the MMS is that it is free of geometric
singularities. The MMS model was applied to the calculation of electrostatic solvation free
energies of 26 proteins [20].

1.3

Quantum mechanical continuum models

In most implicit solvent models, the solute is described as a collection of fixed atomic point
charges, which describes molecular polarizations at the atomistic level of resolution. However, it is well-known that charge rearrangement plays an important role in the solvation
process of proteins in various cases [99]. Similar arguments were used to incorporate the
quantum mechanics (QM) description in the classical implicit solvent theory [227, 228, 61].
The resulting QM version of continuum models, called quantum mechanical continuum models [52, 227, 113, 218, 35, 14, 59], offer the possibility of carrying out accurate quantum calculations in solution and near interfaces. Quantum mechanical continuum models provide
a framework to describe the QM effect in solvent analysis [224]. However, this description
is often compromised by the use of a pre-determined solvent-solute interface model. To integrate a continuum model with a QM description, reaction field potential, i.e., the electric
12

field induced by the polarized solvent, has been introduced as a unifying concept. It is obtained from the electrostatic computation in the framework of continuum models. It also
exists in the Hamiltonian of the solute in the quantum calculation [224, 237, 42]. Therefore,
the quantum formulation of the continuum model involves two problems: (1) the classical
electrostatic problem of determining the solvent reaction field potential with the quantum
mechanically calculated charge density; (2) the quantum mechanical problem of calculating the electron charge density with fixed nucleus charges in the presence of the reaction
field potential. These two problems need to be resolved simultaneously. To carry out these
computations, a intuitive self-consistent iterative procedure can be constructed to resolve
the quantum problem for electron distribution and the classic electrostatic problem for the
reaction field potential [224, 99, 227, 237, 42].

After computing the QM charge density, there are still two ways to implement the solvation analysis. The first approach is to use the continuous QM charge density directly
in the Poisson-Boltzmann equation. The second approach is to fit the QM charge density
into the atomic point charges, and then use the point charges as the source term in the
Poisson-Boltzmann equation. Various schemes have been proposed to compute atomic partial charges with certain efficiency and convenience. The simplest way for atomic partial
charge assignments is the Mulliken analysis method [157]. In this approach, the charge
is distributed according to the orbital occupation. Many other schemes have also been
proposed, including the natural bond orbital analysis, the distributed multipole analysis
(DMA), the wavefunction mapping ‘Class IV’ model, the electrostatic potential expansion
and analysis, etc [184, 214, 202]. Presently, the most widely used method for estimating
atomic partial charges is the least-squared electrostatic potential (ESP ) fitting approach.
13

It was first proposed by Momany and has subsequently been implemented in different ways
with different choices of grid points where the electrostatic potentials are calculated [154].
Examples of such potential-based methods are CHELP, CHELPG, and the Merz-Kollman
scheme [60, 205, 28, 53]. Hu and Yang have recently developed a new object function to
improve the quality and especially the numerical stability of the ESP fitting [110]. ESP
fitting methods are not only widely used in the simulation with molecular mechanical (MM)
force fields, but also extended to the QM/MM simulations as well as the molecular polarization calculation [110]. However, there are some well-known deficiencies in the atomic partial
charge approaches [201, 110]. First, atomic partial charge is not observable, i.e., it can not be
definitely determined by experimental data or directly obtained from quantum calculations.
Therefore, it is a term lacking a rigorous and consistent definition [110]. Additionally, the
approximation of quantum mechanical electron-electron interactions by simple Coulombic
interactions between atomic partial charges leads to inaccurate calculations. Moreover, results from different methods or definitions may show different numerical dependences upon
the QM level of the theory, basis sets used, and the choices of the number and location of
grid points. Finally, there is a concern about the transferability of the atomic partial charges
in different molecules. To avoid these problems, the direct use of the quantum charge density
in the continuum dielectric theory was proposed [227, 237].

The quantum mechanical problem of determining the electron charge density was solved
originally limited to the Hartree-Fock level, which is a traditional way to obtain complicated
many-electron wavefunctions. Density functional theory (DFT) was proposed in 1960s to
provide the ground state of a many-electron system in terms of a single electron wavefunction
[106, 122]. In DFT, the Kohn-Sham equation is the Schr¨dinger equation of a fictitious
o
14

system. It has been more and more popular for quantum calculations in solid state physics
since the 1970s due to its low computational cost when compared with traditional approaches.
Moreover, the results of DFT calculations have been considered accurate enough especially
from 1990s when approximations used in the theory were greatly refined to better model
the exchange and correlation interactions. [127, 21, 33]. DFT is now a leading method
for electronic structure calculations in chemistry, physics and nano-technology. Therefore,
the incorporation of DFT to continuum solvation methods becomes a routine approach in
methods with the QM description of solute and the continuum description of solvent [227].

1.4

Limitations of current models

Current two-scale implicit solvation models have a severe limitation that undermines their
performance in practical applications. While traditional surface definitions have found much
success in biomolecular modeling and computation [212, 139, 62, 123, 25, 73, 114, 136], they
are simply ad hoc divisions of the solute and solvent regions of the problem domain. In
reality, the solvation is a physical process and its equilibrium state should be determined by
fundamental laws of physics. Moreover, these surface definitions confront many challenges.
First of all, as mentioned earlier, from the fundamental physical point of view, macroscopic
properties should vary continuously. Any description of the permittivity changing instantaneously from one point to another is incorrect. Secondly, they admit non-smooth interfaces,
i.e., cusps and self-intersecting surfaces, that lead to well-known instability in molecular
simulations due to extreme sensitivity to atomic positions, radii, etc [188]. Thirdly, each
pre-determined surface definition has its own limitations. For example, the van der Waals
surface (VDW) is not smooth in space. And there are a lot of un-physical solvent pock15

ets inside the solute which cause the fluctuation of the electrostatic field [141]. the widely
used molecular surface (MS) accompanying with the discontinuous dielectric function is not
smooth in time for the molecular dynamic due to its definition, while it is much smoother
in space than VDW and embodies the ratio between contact surface and reentry surface
which maybe an important information of surface roughness. The solvent accessible surface
(SAS) is not suitable in terms of solvation energies, which are the strongest validation of the
PB theory. These difficulties associated with traditional discontinuous dielectric functions
often drive the use of alternative “smoothed” solvent-solute interface definitions [112, 101] by
applying overlapping atom-centered Gaussian or polynomial functions. However, smoothed
interface definitions increase computational cost [70, 72]. Moreover, interatomic crevices
and buried pockets of high dielectric, which are too small for a solvent molecule to occupy,
are introduced. Furthermore, they often overestimate the electrostatic solvation free energy
[217]. Finally, the wide range of surface definitions has often led to confusion and misuse
of parameter (radii) sets developed for implicit solvent calculations with specific surface
definitions.

It is celebrated that the construction of partial differential equation (PDE) based molecular surface by Wei el al. in 2005 [245] generates well defined molecular surfaces for both
small molecules and large proteins [245]. Moreover, the construction of minimal molecular
surface (MMS) is determined by the minimization of the free energy of a macromolecule in
the aquatic environment, and it is free of geometric singularities [20]. However, the MMS,
which incorporates only the minimization of the free energy associated with the surface tension, offers only an approximation to the true physical boundary of a biomolecule in solvent.
Therefore, to account for other important effects that determine the solvent-solute inter16

face, new potential driven geometric flows (PDGFs) needs to be proposed that allow the
incorporation of many other potential effects in surface formation and evolution.
Another criticism of implicit solvent models is the lack of uniqueness in polar and nonpolar
decomposition of the solvation process [148] and the neglect of the polar-nonpolar coupling as
well as solvent-solute interactions [4, 32, 38, 55, 74, 85, 87]. Dzubiella et al [74, 75] considered
this problem by adding a solvent-solute coupling (interaction) term to the total free energy
functional discussed by Sharp and Honig [200] and Gilson et al [97]. A feature of this new
model is that surface tension energy and mechanical work of immersing a molecule into the
solvent were also included in the total free energy functional. However, their initial work
does not provide a protocol for the construction of molecular interfaces and a systematical
analysis of solvation energy for macromolecules. Recently, Cheng et al. [48] have extracted
solvent-solute interfaces from the free energy functional of Dzubiella et al [74, 75] in a setting
very similar to our earlier Eulerian geometric PDE approaches of biomolecular surfaces and
solvation analysis [245, 18, 19, 20].
In the earlier PDE based molecular surface models [245, 20, 74] the solute is described as
a collection of fixed atomic point charges, which, together with the charge described in the
continuum approximation of the surrounding medium, give rise to the total charge source
for the Poisson-Boltzmann equation. The atomic partial charges describe molecular polarizations at the atomistic level of resolution. This approach is able to evaluate many physical
and chemical properties. However, it can not cover the whole range of properties of interest.
In particular, the charge rearrangement in the solute molecule during the transfer from the
gas phase to solution has not been taken into account in the calculation of solvation free
energies. Consequently, the highly accurate analysis of solvent-solute surfaces is discounted
17

by the estimation of charge rearrangement during the solution process. Additionally, those
earlier solvation models depend on parameters from the existing molecular mechanical force
field parameters [147, 12, 112, 116], which are typically parameterized for certain class of
(macro-) molecular systems and may not be appropriate for other class of molecules. Therefore, it is desired to develop a quantum mechanical (QM) description of the solute molecule
in PDE based molecular surface models, which gives accurate, self-consistent and force field
independent charge arrangement treatment during the solavtion process.

1.5

Mathematical issues and numerical challenges

Based on previous discussions, it is clear that there are various mathematical issues involved
in the modeling of solvation. In this work, the solvent-solute boundary can be modeled in the
framework of the differential geometry theory of surfaces and manifolds, which is employed to
result in new coupled geometric and potential flows for the generation of a physical solventsolute boundary and the optimization of solvation energy. Technically, the smoothness of
the resulting solute-solvent boundary is ensured by coupled geometric and potential flows
of parabolic type. A variational framework is established to couple different parts of the
solvation contributions. Governing equations are derived by variational principles. Other
than the modeling strategy, implementations of models encounter many numerical challenges,
which have attracted great mathematical interests for the past several decades. For instance,
the multidomain and multiscale treatment of both systems results in discontinous coefficient
based interface problems and a singular source in the partial differential equation. Highly
accurate and efficient numerical schemes are desired to handle these singularities in the
application of biological systems. Additionally, numerical convergence and efficiency of the
18

self-consistent iteration need to be explored for the derived coupled governing equations. The
involved mathematical issues and associated numerical challenges in model implementations
are outlined as follows:

1.5.1

Geometry, PDE and interface

In this thesis, we consider the solvent-solute boundary as a two-dimensional (2D) differential manifold embedded into a 3D Euclidean space, or a hypersurface, in a Riemannian
manifold. The differential geometry theory of surfaces and manifolds is employed. The
above-mentioned minimal molecular surface, which incorporates only the minimization of
the free energy associated with the surface tension, offers only an approximation to the true
physical boundary of a biomolecule in solvent. To account for other important effects that
determine the solvent-solute interface, we have recently proposed a framework of potential
driven geometric flows (PDGFs) that allow the incorporation of many other potential effects
in surface formation and evolution [17]. The PDGFs are inherently multiscale in nature, and
enable the incorporation of microscopic interactions, such as van der Waals potentials, into
the macroscopic curvature evolution.
From a mathematical point of view, the molecular surfaces are constructed from the
geometric partial differential equation (GPDE) in this thesis. In general, GPDE is a PDE
which controls the motion of curves or surfaces and is merely formulated by the geometric
measure theory. It is related to geometric analysis, manifold theory, topology, PDE, calculus
of variation, and geometric measure theory. GPDE can be applied to the motion of interfaces
problems in the physical or chemistry field, e.g., dissolution, combustion, erosion in the
biology field, biomembrane vesicle problem, and the construction of protein surface. GPDE is
19

also applied to computational geometry, computer graphics, image processing edge detection,
noise removal,image restoration. Moreover, it can be used to obtain some optimal properties
such as surface area minimization, total energy minimization, etc.
The surfaces generated by GPDE possess some attractive features such as smoothness and
a clear geometric sense. GPDE can be constructed by an energy functional based variational
approach. In this method, an energy functional with a physical target is formed. Then, the
first order variation of the energy functional gives rise to geometric PDEs. In the framework
of variational approach, many famous geometric flow equations are derived such as the mean
curvature flow, the Willmore flow, etc.
Because it is difficult to gain explicit solutions of the GPDEs , numerical solutions are
necessary. Numerical solutions of Geometric PDEs can be obtained by the generalized finite
difference method, the finite element methods and the level set like methods.

1.5.2

Geometric flow equation

Much of the recent development in implicit solvent models is due to the use of geometric
flows [245, 18, 20, 17, 243], particularly mean curvature flows, which have been of considerable interest in applied mathematics for decades [172, 187, 197, 84, 100, 153, 170, 190, 192,
197, 208, 247]. Earlier research work and part of present research are focused on image processing [172, 187], computer vision, materials design [197] and surface smoothing [249, 256].
Computational techniques using the level set theory were devised by Osher and Sethian
[172, 187, 197] and have been further developed and applied by many others [37, 54, 206].
An alternative approach for image analysis is to minimize a functional in the framework of
the Mumford-Shah variational functional [158], and/or the Euler-Lagrange formulation of
20

variation [30, 36, 134, 171, 187, 189]. Wei introduced some of the first family of high-order
geometric flow equations for image analysis [242]. In fact, the nonlinear production term
in these high-order operators provides a framework to accommodate the PDGF in our later
formation for macromolecular surfaces. Their high-order geometric flow equations have led
to many interesting applications [242, 244, 215, 144, 96, 39]. Mathematical analysis of these
high order equations in Sobolev space was carried out by Bertozzi and Greer [27, 103, 104],
who proved the existence and uniqueness of the solution to a case with H 1 initial data and
a regularized operator. A similar analysis was performed by Xu and Zhou [250]. Wei and
Jia also introduced a coupled geometric flow equation system for image edge detection [244].
Such an algorithm works extremely well with texture images. Recently, Wei and his collaborators have proposed an evolution operator based single-step method for image denoising
and enhancement [215]. Most recently, a family of differential geometry based multiscale
models has been developed by Wei for chemical and biomolecular systems, including fuel
cells, ion channels, DNA packing, nanofluidic systems, and virus evolution [243]. These
models describe not only the structure, but also the dynamics and transport of the above
mentioned chemical and biomolecular systems.

1.5.3

Highly accurate and efficient solver for interface problems

In general, electrostatic energy is much larger than the non-electrostatic part so that the accuracy of electrostatic potential calculation based on the Poisson-Boltzmann (PB) equation
plays a critical role in controlling the accuracy of the total solvation free energy. Therefore,
numerical methods that are able to deliver highly accurate solution of the PB equation is
desirable. In the present Lagrangian model, there exists a sharp solvent-solute interface and
21

it leads to a discontinuous dielectric constant definition in the PB equation. When the discontinuous dielectric profile is applied across the interface, the standard numerical methods,
including the centered finite differences scheme, lose their accuracy and convergent order.
This problem is aggravated by complex geometric shapes, possible geometric singularity, and
singular charges of biomolecules. In the worst-case scenario, the standard numerical methods
do not converge at all for complex irregular solvent-solute interfaces [252, 93].
The solution of elliptic equations with discontinuous coefficients and singular sources is a
challenging problem in computational mathematics. In order to achieve high-order numerical
accuracy, it is indispensable to develop mathematical interface techniques. Peskin pioneered
the immersed boundary method (IBM) [124, 176] to address this class of problems. Recently,
many other elegant methods have been proposed, including the ghost fluid method [79, 137],
the upwinding embedded boundary method [34], finite-volume-based methods [164], and
integral equation methods [151]. A major advance in the field was due to LeVeque and
Li [129], who proposed a remarkable second order sharp interface scheme, the immersed
interface method (IIM) [129, 135]. Chen and Strain discussed a piecewise-polynomial discretization and Krylov-accelerated multigrid for elliptic interface problems [44]. However,
these interface techniques have not been implemented for the Poisson-Boltzmann equation
in the context of realistic biomolecules.
Wei and his coworkers have recently proposed a highly accurate algorithm, the matched
interface and boundary (MIB) method [254, 253, 258, 261, 260] for solving elliptic equations.
Many essential ideas of the current MIB method were introduced in earlier interface schemes
for solving Maxwell’s equations [258]. The MIB is of arbitrarily high-order accuracy in
principle, and sixth-order accurate MIB schemes have been constructed [253, 261]. Wei’s
22

group has developed three generations of MIB based PB solvers, MIBPB-I [259], MIBPB-II
[252], and MIBPB-III [93]. The MIBPB-I is the first PB solver that explicitly enforces the flux
continuity conditions at the dielectric interface in a biomolecular context; however, it cannot
maintain its designed order of accuracy in the presence of molecular surface singularities,
such as cusps and self-intersecting surfaces commonly occurred in biomolecular systems [188].
This problem was first addressed in the MIBPB-II by utilizing an advanced MIB technique
developed by Yu et al. [253]; however, the MIBPB-II still loses its accuracy when the mesh
size is as large as half of the smallest van der Waals radius, because of the interference of the
interface and singular charges. To split the singular charge part of the solution [262, 43, 31],
a Dirichlet to Neumann mapping approach [51] was designed in the MIBPB-III, which is
by far the most accurate and stable PB solver. To our knowledge, the MIBPB method is
the only existing method that is able to offer second order accuracy in solving the PoissonBoltzmann equation with discontinuous coefficients, singular sources and arbitrarily complex
interfaces. The MIBPB is a few orders of magnitude more accurate at a given mesh size and
about three times faster at a given accuracy than some traditional PB solvers [93].
The most important idea in all interface techniques is to take care of interface conditions,
which may vary from systems to systems. Complex interface conditions are needed for
the Helmholtz equation [257] and Maxwell’s equations [258]. For the Poisson-Boltzmann
equation, interface conditions are the following

[φ]Γ = φ+ (r) − φ− (r) = 0
[ǫφ]Γ = (ǫs ∇φ+ ) · N − (ǫm ∇φ− ) · N = 0.
23

(1.3)

where φ+ and φ− are the electrostatic potential inside and outside the solvent-solute surface,
respectively. Different methods may have different strategies in dealing with these conditions.
The MIB method has a unique set of procedures in implementing Eq. (1.3). The interested
reader is referred to earlier work [254, 253, 258, 261, 260].
In this work, we make use of MIBPB-III scheme for PDE based interface problems. We
take dielectric constants ǫm = 1 and ǫs = 80 in our calculations. We use the Dirichlet farfield boundary condition and the electrostatic potential values at the boundary are practically
obtained by the sum of potential contributions from individual atomic charges with an
exponential decay factor [93]. The MIBPB-III is used to handle discontinuous dielectric
constants, complex geometry and charge singularity. Note that although the geometry is
complex, there is no geometric singularities, such as cusps and intersecting surfaces, in the
biomolecular surfaces generated by our approaches. The extraction of surface information is
carried out by the marching cubes algorithm embedded in our codes.

1.5.4

Self-consistent iterative methods

It will be seen , through the optimization of the solvation energy, that the resulted generalized Poisson-Boltzmann (PB) equation and the generalized potential driven geometric flow
equation are fully coupled. The optimized electrostatic potential is obtained by solving the
Poisson-Boltzmann equation in which solvent-solute interface is used to determine the dielectric constant and the domain decomposition. The interface is generated by the solution of
the potential driven geometric flow equation, which in turn depends on the electrostatic potential. In other words, the solution of the potential driven geometric flow equation requires
the knowledge of electrostatic potential, while the solution of the generalized PB equation
24

requires the input of the interface definition.
Therefore, the coupled generalized geometric flow equation and the generalized PB equation need to be solved simultaneously, in the present differential geometry based solvation
model. The existence and the uniqueness of their solution, under the biomolecular context,
can be an interesting mathematical issue. Numerically, both the convergence and efficiency
of the solutions of the coupled system will be given quite a bit of attentions.
In practice, this coupled nonlinear system can be solved by an iterative procedure, until a self-consistency is reached. Iterative methods are often the only choice for nonlinear
equations. In computational mathematics, an iterative method is a mathematical procedure
that leads to approximate solutions for a class of problems. The most common iterative
method is Newton’s method. Other well-known examples include the Gummel’s method,
the steepest descent method, and the conjugate iterative method. An iterative method is
called convergent if the corresponding solution sequence converges for given initial approximations. A mathematically rigorous convergence analysis of an iterative method need to be
performed; however, heuristic-based iterative methods are also commonly used.

1.6

The rest of this thesis

The rest of this thesis is organized as follows. In Chapter 2, we present the Eulerian formulation of our differential geometry based solvation models. The Eulerian analysis of
biomolecular surfaces utilizes the well-known coarea theorem from the geometric measure
theory. The resulting operator from surface area minimization can be identified as a generalized Laplace-Beltrami operator in a 3D dimension. Chapter 4 is devoted to the Lagrangian
formulation. Lagrangian analysis of biomolecular surfaces makes the direct use of differential
25

geometry theory of surfaces and manifolds. The surface minimization leads to the LaplaceBeltrami operator defined in 2D domain, or the mean curvature operator. In Chapter 4, the
connection of two representations is analyzed in the present work. The structure of governing
equations, and the accuracy and efficiency of two formulations are compared. The objective
of Chapter 3 is to incorporate a quantum mechanical description of charge density into our
earlier differential geometry based solvation model, which is described in Chapter 2. This
thesis is concluded by discussing the main achievements and future work.

26

Chapter 2
Eulerian formulation
This chapter presents a differential geometry based model for the analysis and computation
of the equilibrium property of solvation. Differential geometry theory of surfaces is utilized
to define and construct smooth interfaces with good stability and differentiability for use in
characterizing the solvent-solute boundaries and in generating continuous dielectric functions
across the computational domain. A total free energy functional is constructed to couple
polar and nonpolar contributions to the solvation process. Geometric measure theory is employed to rigorously convert a Lagrangian formulation of the surface energy into an Eulerian
formulation so as to bring all energy terms on an equal footing. By minimizing the total free
energy functional, we derive coupled generalized Poisson-Boltzmann equation (GPBE) and
generalized geometric flow equation (GGFE) for the electrostatic potential and the construction of realistic solvent-solute boundaries, respectively. By solving the coupled GPBE and
GGFE, we obtain the electrostatic potential, the solvent-solute boundary profile, and the
smooth dielectric function, and thereby improve the accuracy and stability of implicit solvation calculations. We also design efficient second order numerical schemes for the solution of
27

the GPBE and GGFE. Matrix resulted from the discretization of the GPBE is accelerated
with appropriate preconditioners. An alternative direct implicit (ADI) scheme is designed to
improve the stability of solving the GGFE. Two iterative approaches are designed to solve
the coupled system of nonlinear partial differential equations. Extensive numerical experiments are designed to validate the present theoretical model, test computational methods,
and optimize numerical algorithms. Example solvation analysis of both small compounds
and proteins are carried out to further demonstrate the accuracy, stability, efficiency and
robustness of the present new model and numerical approaches. Comparison is given to
both experimental and theoretical results in the literature.

This chapter is organized as follows. Section 2.1 is devoted to the theoretical foundation
of the present differential geometry based solvation model. A variational framework is established to couple different parts of the solvation contributions. Governing equations are
derived by variational principles. The solution of the governing equations leads to physical solvent-solute boundaries and accurate solvation free energies. Numerical methods and
algorithms are discussed in Section 2.2. Schemes of the second order numerical accuracy
are designed for the construction and evolution of solute characteristic function. Appropriate preconditioners are used for solving the generalized Poisson-Boltzmann equations. The
coupled equations are solved by two iterative schemes. Section 2.3 presents validation and
analysis of the proposed numerical approaches. The accuracy and convergence of various
computational schemes, including the surface area formulation based on the geometric measure theory, are carefully tested to ensure their computational reliability and efficiency. The
applications of the proposed theories, methods and algorithms are considered to two sets
of compounds: small molecules and proteins in Section 2.4. Finally, this chapter ends with
28

1

s

S
1−S

0.5

0
−6

−4

−2

0

x

2

4

6

Figure 2.1: The cross line of S and (1 − S) of a diatomic system described in Section 2.3.3

concluding remarks.

2.1

Theory and model

In this section, a differential geometry based model of solvation is briefly described for macromolecules and their aquatic environment that are near equilibrium. More details about the
differential geometry based multiscale formalism, particularly dynamics and transport aspects, can be found elsewhere [243]. For a system near equilibrium, the density of charged
particles in the solvent can be approximated by the Boltzmann distribution, which considerably reduces the number of degrees of the freedom of the solvation system. Alternatively,
the Nernst-Planck equations or the full set of the Navier-Stokes equations might be utilized
to describe systems that are far from the equilibrium [243].
29

2.1.1

Solute-solvent boundary

Let us consider a multi-domain setting of a macromolecule and solvent system. The macromolecule is described in discrete atomic detail, while the aqueous solvent is treated as a
continuum. Therefore, the domain Ω ∈ R3 is essentially divided into two (types of ) regions, i.e., aqueous solvent domain Ωs and macromolecular domain Ωm . Therefore, one has
Ω = Ωs

Ωm . However, because electron wavefunctions of the solvent and the solute overlap

at the atomic scale, Ωs and Ωm should overlap with each other at the boundary of molecules
and solvent, i.e., Ωb = Ωs

Ωm = ⊘, where Ωb is the region of solvent-solute boundary.

Therefore, we propose a characteristic function S : R3 → R to characterize this overlapping
solvent-solute boundary. As such, S(x) is a description function or a characteristic function
of the solute domain, i.e., it is one (S = 1) inside the biomolecule and zero (S = 0) in
the aquatic solvent. However, S takes a value between zero and one at the solvent-solute
boundary region. Therefore, (1 − S) is a description function or a characteristic function for
the solvent domain. The profiles of S and (1 − S) are depicted in Figure 2.1 for a simple
system. It is seen that there is a transition region at the solvent-solute boundary where the
solvent and the solute regions overlap. Obviously, in our model, the evaluation of all the
solvent-solute properties depends on S. Physically, S and thus the profile of solvent-solute
boundary, must be determined by the energy minimization principle. Therefore, our task
is to identify the energy functional that to be optimized. This task is accomplished via the
differential geometry of surfaces and manifolds in the present work.
30

2.1.2

Total free energy functional

The solvation process of macromolecules involves a number of interactions. As discussed in
the Introduction, typically, the free energy of solvation models consists of polar and nonpolar
contributions, as well as polar and nonpolar interactions.

2.1.2.1

Polar free energy functional

The polar part is standardly represented by electrostatic interactions, which are of special
importance because of their long range and influence on polar or charged molecules including
water, aqueous ions, and amino or nucleic acids. They are also some of the most important
aspects that determine the physical and chemical properties of biomolecules, such as protein
folding, protein-DNA binding, gene expression and regulation, etc. The widely used free
energy functional of the electrostatic system was given by Sharp and Honig [199] and Gilson
et al [97]. However, their formulation is based on a sharp interface that separates the
solvent and solute domains. In our formulation, we incorporate the function S into the polar
solvation free energy functional

Gp =

1
S ρm φ − ǫm |∇φ|2
2
Ω


Nc

1
+(1 − S) − ǫs |∇φ|2 − kB T
n0 (e−φQi /kB T − 1) dr
i

2
i=1

(2.1)

where φ is the electrostatic potential whose domain is the whole computational domain Ω,
and ǫs and ǫm are the dielectric constants of the solvent and solute, respectively. Here
ρm =

j qj δ(r − xj ) is the density of molecular charges, with qj being the partial charge

on an atom located at xj , Qi is the charge of ion species i, Nc is the number of ion species,
31

kB is the Boltzmann constant, T is the temperature, and n0 is the bulk concentration of the
i
ith ionic species. The term associated with S is the electrostatic free energy of the solute
and that with (1 − S) is the electrostatic free energy of the solvent.
The above electrostatic free energy functional is inherently multidomain in nature and
the domain is divided into the solute subdomain and the solvent subdomain as indicated by
S and 1−S, respectively. These subdomains do not have to be mutually exclusive. A discrete
description of the solute and a continuum description of the solvent are also employed in Eq.
(2.1) in the framework of the implicit solvent treatment, in which the charge density of mobile
ions follows the Boltzmann distribution. Moreover, it will be demonstrated that the present
electrostatic free energy functional is able to reproduce the classical Poisson-Boltzmann
equation when a sharp solvent-solute interface is used and S becomes a Heaviside function.
Finally, we note that the terms that are quadratic in the potential gradient in Eq. (2.1) have
negative signs. Therefore, the free energy will be optimized instead of being minimized. In
this work, we have adopted the earlier sign convention in the field [199, 97].

2.1.2.2

Non-polar free energy functional

For the nonpolar contribution, we consider the nonpolar solvation free energy functional
discussed in the Introduction.

Gnp = γ · Area + p · Vol + ρ0

Ωs

U att dr

(2.2)

To obtain a functional relation for S, it is necessary to rewrite nonpolar free energy
32

formulation in terms of S(r). The enclosed volume of biomolecule can be given by

Vol =

Ωm

dr =

Ω

S(r)dr.

(2.3)

Similarly the attractive dispersion term can be rewritten in the form

ρ0

Ωs

U att dr = ρ0

Ω

(1 − S(r))U att dr,

(2.4)

where we assume that the solvent bulk density ρ0 is a constant in space.

Typically, one expresses the area of a unique surface as a surface integration over the
biomolecular boundary in the Lagrangian formulation. However, this approach does not
work directly in our formulation because no sharp solvent-solute boundary is assumed. In
fact, the concept of the surface area cannot be defined in the same manner as in the sharp
surface case. For a smooth boundary, there are infinitely many surfaces and the surface area
can be defined as a weighted mean of this family of surfaces. Additionally, for practical
purpose, we need an appropriate Eulerian formulation for the surface area so that we can
put all energy contributions on an equal footing. Therefore, we need to convert the surface
integral into a volume one. To this end, we make use of the coarea formula in the geometric
measure theory [78]. For a scalar field B in R3 , with C 1 continuity condition, integrating a
function f over its isolevel c in a region Ω can be done directly by a volume integral over Ω
through the expression

R B −1

Ω

f dσdc =
33

Ω

∇B f (r)dr,

(2.5)

where c denotes an isovalue of B, and B −1 represents the c-isosurface, i.e., the set of points
{rc } such that B(rc ) = c. Here, the coarea formula prescribes a relationship between
the sum of area integrals and a global volume integral. In our case, we introduce the
concept of mean surface area of the family of isosurfaces which are subsets of point satisfying
{S(r) = y}, where 0 ≤ y ≤ 1. Therefore the mean surface area can be given by a volume
integral as

Area =
=

1
0

S −1 (c)

Ω

dσdc

(2.6)

∇S(r) dr.

Ω

Note that ∇S = 0 only in the region of the solvent-solute boundary. Numerical test of this
formulation will be presented in Section 2.3.1.

Finally, the electrostatic free energy functional is complemented by the nonpolar free
energy functional to give the total free energy functional of solvation for biomolecules at
equilibrium

Gtotal =

Ω

γ ∇S(r) + pS(r) + ρ0 (1 − S(r))U att

(2.7)

1
+ S(r) ρm φ − ǫm |∇φ|2
2


Nc
1
+ (1 − S(r)) − ǫs |∇φ|2 − kB T
n0 e−φQi /kB T − 1  dr.
i
2
i=1
Note that the polar and nonpolar parts are coupled via the characteristic function S, which is
determined by the total energy optimization instead of the surface free energy optimization
as done in our earlier work [20]. The above total free energy expression provides a basis for
34

the evaluation of the solvation free energy and a starting point for the derivation of governing
equations for the solvation analysis. A similar coupling of polar and nonpolar interactions
was described previously by Dzubiella and co-workers [74, 75]; however, the implementation
of non-polar interactions and the representation of continuum and discrete domains differ
significantly from the present work.

2.1.3

Governing equations

The solvation free energy functional is a functional in terms of characteristic function S and
potential φ. The integral is taken over the whole space. From the physical point of view, there
should exist an optimal function S(r) and an optimal potential φ at the equilibrium state
which optimize the total energy. Since S and φ can vary independently in our formulation,
to optimize Gtotal , it is necessary that
δGtotal
⇒ Sρm + ∇ · ([(1 − S)ǫs + Sǫm ]∇φ) + (1 − S)
δφ

Nc
i=1

n0 Qi e−φQi /kB T = 0 (2.8)
i

and
δGtotal
∇S
⇒ −∇ · γ
δS
∇S

1
+ p − ρ0 U att + ρm φ − ǫm |∇φ|2
2
Nc
1
2+k T
+ ǫs |∇φ|
n0 (e−φQi /kB T − 1) = 0,
B
i
2
i=1

where ∇ · γ ∇S
∇S

(2.9)

is a generalized Laplace-Beltrami operator, which is a generalization of

the usual Laplacian operator to a smooth manifold [17, 243]. In general, γ can be a function
of the position γ = γ(r) to reflect surface hydrophobicity at different locations. However,
35

Epsilon

80
60
40
20
0
−6

−4

−2

0

x

2

4

6

Figure 2.2: The cross line profile of ǫ(S) of a diatomic system described in Section 2.3.3.
Here, we have set ǫs = 80 and ǫm = 1.
it is treated as a constant in our present computation. From Eq. (2.8) we result in the
generalized Poisson-Boltzmann equation (GPBE)
Nc
−∇ · (ǫ(S)∇φ) = Sρm + (1 − S)

i=1

n0 Qi e−φQi /kB T ,
i

(2.10)

where the dielectric function is given by

ǫ(S) = (1 − S)ǫs + Sǫm .

(2.11)

This expression provides a smooth dielectric profile. Figure 2.2 shows the cross line of the
dielectric function ǫ(S) of a diatomic system. It is seen that there is a smooth transition
region for the dielectric constant to change from ǫs to ǫm . The solution procedure of Eq.
(2.10) can differ very much from that of the standard PB equation, due to the smooth dielectric function. Particularly, many mathematical difficulties of solving elliptic equations with
discontinuous coefficients [258, 261, 260, 254, 253] can be avoided in the present generalized
Poisson-Boltzmann equation.
36

For a weak electrostatic potential, i.e., φ ≪ 1, one can linearize the generalized PB
equation
−∇ · (ǫ(S)∇φ) + (1 − S)κ2 φ = Sρm ,

(2.12)

u
where κ is a modified Debye-H¨ckel screening function describing the ion strength [108].
Furthermore, the solution of Eq. (2.9) leads to a “physical solvent-solute boundary”
(S). As discussed in earlier work [20, 17, 243], the solution of this elliptic partial differential
equation can be attained via a parabolic partial differential equation

∂S
=
∂t

∇S

∇· γ

∇S
∇S

+V ,

(2.13)

where the generalized “potential” V is defined as

1
1
V = −p+ρ0 U att −ρm φ+ ǫm |∇φ|2 − ǫs |∇φ|2 −kB T
2
2

Nc
i=1

n0 e−φQi /kB T − 1 . (2.14)
i

Note that Eq. (2.13) has the same differential operator as the mean curvature flow equation
[20], except for the extra external source term. Therefore, it is a special case of the potential
driven geometric flow equation proposed in our earlier work [17]. In Eq. (2.13), as t → ∞,
the initial profile of S evolutes into a steady state solution, which solves the original Eq.
(2.9).
It is interesting to see that the sharp solvent-solute interface and the standard PB equation, as well as related interface conditions, can be recovered from Eq. (2.10). For a sharp
interface, S becomes a Heaviside function, having value 1 for the solute subdomain and 0
for the solvent subdomain. As such, the smooth transition region in the dielectric function
disappears and the dielectric function becomes discontinuous. Then, Eq. (2.10) reduces to
37

the classical form of the Poisson-Boltzmann equation [108]
−ǫm ∇2 φm = ρm
−ǫs ∇2 φs =

∀r ∈ Ωm

(−φs Qj /kB T )
∀r ∈ Ωs
j Qj cj e

with appropriate interface conditions

φs = φm , and

ǫm ∇φm · n = ǫs ∇φs · n ∀r ∈ Γ,

(2.15)

where φm and φs represent the potential in the solute domain Ωm and solvent domain Ωs ,
respectively, Γ denotes the sharp interface, and n is the normal vector of the solvent-solute
sharp interface.
Note that the generalized Poisson-Boltzmann Eq. (2.10) and the potential driven geometric flow equation (2.13) are strongly coupled. Therefore, these two equations have to be
solved by appropriate iterative procedures. This aspect will be discussed in Section 2.2.3.

2.2

Methods and algorithms

This section presents a variety of computational methods and algorithms for the solution of
the generalized Poisson-Boltzmann equation and the generalized geometric flow equation.

2.2.1

Discretization schemes of the governing equations

We design second order finite difference schemes for governing equation derived from the free
energy optimization.
38

2.2.1.1

The generalized Poisson-Boltzmann equation

For the solution of the generalized PB equation, the finite difference scheme is utilized in this
study. The continuous dielectric definition allows us to obtain accurate solution by using the
standard second order center difference scheme. Let the pixel (i, j, k) represent the position
(xi , yj , zk ). The discretized form of Eq. (2.12) is

+
+
+
+
+
=

1
ǫ(xi + h, yj , zk )[φ(i + 1, j, k) − φ(i, j, k)]
2
1
ǫ(xi − h, yj , zk )[φ(i − 1, j, k) − φ(i, j, k)]
2
1
ǫ(xi , yj + h, zk )[φ(i, j + 1, k) − φ(i, j, k)]
2
1
ǫ(xi , yj − h, zk )[φ(i, j − 1, k) − φ(i, j, k)]
2
1
ǫ(xi , yj , zk + h)[φ(i, j, k + 1) − φ(i, j, k)]
2
1
ǫ(xi , yj , zk − h)[φ(i, j, k − 1) − φ(i, j, k)]
2
(1 − S(i, j, k))κ2 φ(i, j, k)h2 − S(i, j, k)q(i, j, k)/h

(2.16)

where h is the grid spacing, and q(i, j, k) is the fractional charge at grid point (xi , yj , zk ),
which is resulted from the interpolation of the charge density ρm . The second order interpolation (i.e., the trilinear mapping) is used to distribute charges. Thus, the discretized PB
equation can be cast into the standard linear algebraic equation system of the form AX = B,
where X is the solution of the interest, A is the discretization matrix and B is the source
term associated with the continuum and discrete charges. The boundary condition is built
up by the far field condition and practically obtained by the sum of potential contributions
from individual atomic charges with a decay factor from the continuum charge strength κ
[93]. We have explored the use of the biconjugate gradient method as the linear solver.
39

Matrix acceleration is discussed in a later section. The initial guess of the solution is set to
0 and the convergence tolerance is set as 10−6 . It is shown in the test section that the PB
solver is able to deliver the designed second-order accuracy. The solution of geometry flow
equation (2.13) is described in detail in Appendix A.

2.2.2

Acceleration procedures

The computational efficiency of the solution process is an important issue and can be a
bottleneck for further applications of the present model. Particularly, when this model is
applied to molecular dynamic simulation, the generalized PB equation and geometry flow
equation are to be solved up to millions of times. Therefore, any nontrivial improvement
in computational efficiency will make the present model more feasible to many practical
applications in chemical and biological systems.

2.2.2.1

Precondition of the PB solver

The linear algebraic system of the discretized PB equation can be solved by two major
approaches: direct methods and iterative methods. Large memory requirement prohibits
direct methods to be used in the matrix resulted from large chemical and biological systems. Widely used iterative methods, including Gauss-Seidel and successive over-relaxation
(SOR), work well for the generalized PB equation, but typically converge slowly. Conjugate Gradient method is quite efficient for symmetric and positively definite matrices.
However, the sparse matrix A resulted from Eq. (2.16) is seven-fold banded but nonsymmetric because the dielectric distribution function is not a constant and varies in the
transition region.

The biconjugate gradient (BiCG) method can be a good choice for
40

non-symmetric systems and has been adopted in a variety of our MIB schemes [258, 261,
260, 254, 253], but attentions are still to be paid in regard to the convergence issue. We
have studied the application of pre-conditioners in two linear solver libraries, the SLATEC
(http://people.sc.fsu.edu/˜burkardt/f src/slatec/slatec.html) and the PETSc (http:// www
.mcs.anl.gov/petsc/petsc-as/) to the solution of the PB equation [40]. It turned out that
combination of stabilized biconjugate gradient method (BiCG) and the blocked Jacobi preconditioner (BJAC) from the PETSc and the combination of the orthomin method (OM) and
the incomplete LU factorization preconditioner (ILU) from the SLATEC performed better
compared to other tested solvers, preconditoners and their combinations [40]. In this study,
we focus on the combination of the ILU and the OM from the SLATEC, which is easy to incorporate into the present iteration procedure and provides a stand-alone package, while the
PETSc needs to be pre-installed before being used. In Section 2.3, we further demonstrate
the improvement by the combination of pre-conditioners and the iterative linear solvers.

2.2.2.2

Initial guess of the generalized PB solution

A good initial guess is always desired for the speedup of the iterative PB solver. Normally, the
initial guess can be simply set to 0 because it is complicated and computationally expensive
to find good ones. However, in our iteration procedure, it is found that the electrostatic
potential distribution does not change dramatically from the prior calculation due to the
small adjustment in dielectric from the prior step. Therefore, it is beneficial to take the prior
potential as a good guess for solving the linear system. It turns out that the generalized PB
solver converges faster when the electrostatic potential from the previous iteration is used
as an input.
41

2.2.2.3

Convergence criteria in the generalized PB solver

The convergence criterion directly influences the accuracy and CPU cost of the solution of
the generalized PB equation. The smaller convergence criterion, the more accurate the solution of linear system is. However, the smaller convergence criterion requires more iterations
and longer CPU time in the iterative solution process. Therefore, it is worthwhile to find
a criterion which is a good compromise between the accuracy and the efficiency. Typically,
a value of 10−6 is used in many chemical and biological applications. Later on we will
numerically investigate the effect of convergence criterion on the electrostatic solvation energy, mean surface area and mean volume which are used to compute the total solvation free
energy. With the 10−8 as a standard, we will examine the efficiency and the accuracy for
several relaxed convergence criteria, such as 10−6 , 10−4 , 10−3 , 10−2 and 10−1 .

2.2.3

Dynamical coupling of the generalized Poisson Boltzmann
and geometry flow equations

As described in Section 2.1, the present differential geometry based solvation model prescribes
a procedure to set up the total free energy functional of the solvation. By the variational
principle, we obtain generalized coupled PB equation (2.10) and potential driven geometric
flow equation (2.10). The solution of these coupled nonlinear equations provides a “physical”
dielectric profile ǫ(S) and the electrostatic potential φ and thereby enables the calculation of
the solvation free energy. The solution of the potential driven geometric flow equation (2.10)
requires the knowledge of φ, while the solution of the generalized PB equation (2.10) requires
the input of S and ǫ(S). Therefore, in principle, the geometric flow equation needs to be
solved simultaneously with the generalized PB equation until a self-consistency is reached.
42

In this study, we explore two self-consistent iteration procedures.

2.2.3.1

Approach I

The iteration process can be carried out by breaking up the process into an iterative sequence
of two steps as follows: Starting with an initial guess of characteristic function S, one finds
out the temporary electrostatic potential φ by solving the generalized PB equation with a
given initial S. Once the electrostatic potential is obtained, the electrostatic energy can
be calculated. The second step is to solve the time-dependent generalized geometric flow
equation for S with a prior calculated potential φ. In this step, the time integration can
usually reach a quasi-steady state. With the updated quasi-steady S, one can come back
to the first step for the next cycle until the solvation free energy converges to within a
pre-determined criteria. However, in practice, simply re-inserting S into the PB solver may
diverge. Because the quasi-steady S may vary dramatically during the iteration. Note that
all changes in S are concentrated around the solvent-solute boundary, as the final solution
of the potential driven geometric flow equation reflects the balance between the intrinsic
curvature energy and the external potential terms. A large change in S near the solventsolute boundary in turn leads to much adjustment in the electrostatic potential which differs
much in the solute and solvent regions. To avoid this problem, we adopt a simple relaxation
algorithm: the characteristic function S used for the PB solver is a linear combination of
the previous one Sold and the newly generated one Snew

S = αSnew + (1 − α)Sold ,
43

0 < α < 1.

(2.17)

It turns out that the convergence of the generalized PB equation is guaranteed if α is small
enough. In this work, α can be taken in the range from 0.1 to 0.7. The choice of α is
explored later. Note that this approach may fail sometimes when the generalized geometric
flow equation blows up due to a large variation in the temporary electrostatic potential. We
therefore utilize a similar procedure for the electrostatic potential used in the evolution of
the generalized geometric flow equation

φ = α′ φnew + (1 − α′ )φold ,

0 < α′ < 1,

(2.18)

where φold and φnew are previous and newly resolved electrostatic potentials, respectively.
This treatment can avoid the blow-up of the generalized geometric flow solution.

2.2.3.2

Approach II

To reduce the dramatic changes in S and φ, we can consider a straightforward way in
which solving generalized PB equation follows each time-step integration of the generalized
geometry flow equation. However, this treatment makes the whole iterative procedure computationally over expensive as many more PB solution processes are required. Additionally,
it may not be necessary since the change in the S from one time step to another one is
so small that the change in the corresponding potential distribution should be very small.
Indeed, it is practical to update electrostatic potential after a number of time steps (i.e.,
10 to 100 steps) in the generalized geometry flow equation integration rather than every
time step. We call the number of time integration between two φ updatings the number of
intermittency Nstep . This approach effectively speeds up the whole process. Additionally,
the relaxation algorithm given in Eqs. (2.17) and (2.18) can also be used here to guaran44

tee the convergence. Moreover, in this approach, one better starts the iterative process by
solving the S from Eq. (2.13) without the electrostatic potential term. So that the later
iteration procedure can focus on the impact of electrostatic potential to the generation of
the solvent-solute boundary. This treatment reduces the total iteration number and save the
computational time significantly.
In fact, there is a relationship between Approach I and Approach II. When the number
of the time integrations becomes larger and larger, Approach II returns to Approach I. In
Section 2.3, we systematically study the difference between these two approaches. This can
be done by comparing the impacts of different approaches on the resulting total solvation free
energy, surface area and volume of the solute molecule. It is found that these two approaches
lead to the same results. This, to some degree, indicates the reliability and validity of the
proposed iteration procedures.

2.2.4

Evaluation of the solvation free energy

Once the electrostatic potential φ and the characteristic function S are obtained, the solvation free energy is given by
∆G = G − G0

(2.19)

where G0 is the energy calculated from the homogeneous environment with ǫs = ǫm = 1
without the nonpolar energy part. Therefore, we have

∆G = Gp + Gnp − G0 .
45

(2.20)

The expressions of Gp and Gnp are taken from Eq. (2.7). Here Gp − G0 can be considered
as the electrostatic solvation free energy. In all calculations presented here except for salt
effect calculation, mobile ions will be set to zero corresponding to a solution without salt.
Therefore we have the following approximation

Gp =

Ω

S(r)ρm φdr −

1
1
ǫ(S(r))|∇φ|2 dr ≈
S(r)ρm φdr.
2 Ω
2 Ω

(2.21)

Discretizing the integral yields

1
Gp =
2

Nm
Q(ri )φ(ri ),

(2.22)

i=1

where Q(ri ) is the ith partial charge at ri in the biomolecule, and Nm is the total number
of partial charges. Now the electrostatic solvation free energy can be calculated as

1
∆Gp = Gp − G0 =
2

Nm
i=1

Q(ri )(φ(ri ) − φ0 (ri )),

(2.23)

where φ and φ0 are electrostatic potentials in the presence of the solvent and the vacuum,
respectively. Here φ is computed from the generalized Poisson equation (2.10) using the
continuous dielectric distribution

−∇ · (ǫ(S)∇φ(r)) = Sρm

(2.24)

where ǫ(S) and ρm are the same as the ones in Eq. (2.10). The homogeneous solution φ0
is computed with ǫ(S) = ǫp in the whole domain. The nonpolar part, Gnp , is computed
exactly by Eq. (2.2).
46

2.3

Numerical test and validation

This section provides systematic validations for the computational algorithms and schemes
proposed in the last two sections. We first examine the behavior of the coarea formula,
then continue testing through various equation solvers, and finally check the impact of the
potential term in our generalized geometric flow equation.

2.3.1

The behavior of the coarea formula

As described earlier, the coarea formula plays an important role in describing the mean
surface area of an infinite family of smooth solvent-solute boundaries by a volume integral.
This Eulerian formulation puts the free energy of the surface area and other free energies
in an equal footing. Usually, the isovalue of the surface area in the coarea formula can be
any positive real number. But for the present derivation, it is limited to be between 0 and 1
because S is defined as a characteristic function of the solute. Here, we numerically explore
the behavior of the coarea formula in a bounded open set. To this end, we design some test
cases as follows: Let B be a smooth function with a specific expression according to the
geometry in the coarea formula, we set

mean surface area of {x|0 < B(x) < 1} =

1
0

B −1

Ω

dσdc =

Ω

∇B dx

(2.25)

where Ω is a bounded open set. Therefore, the mean surface area has the same value as the
volume of open set Ω {x|0 < B(x) < 1}. Computationally, integrating over the norm of
the gradient of B gives rise to the corresponding mean surface area. The volume integral of
47

a density function f is just simply approximated by

Ω

f (x, y, z)dr ≈

f (xi , yj , zk )h3

(2.26)

(i,j,k)∈J

where the summation is over J, the set of points inside Ω, and (xi , yj , zk ) is the coordinates
of grid points (i, j, k). Table 2.1 lists the numerical results and exact values of the surface
areas for the following cases
(a) A unit sphere: Ω = {(x, y, z)|x2 + y 2 + z 2 ≤ 1} and B =

x2 + y 2 + z 2

(b) A cylinder: Ω = {(x, y, z)|x2 + y 2 < 1, −4 ≤ z ≤ 4} and B =

x2 + y 2

(c) A ellipsoid: Ω = {(x, y, z)|(x/a)2 + (y/b)2 + (z/c)2 ≤ 1} and
B=

(x/a)2 + (y/b)2 + (z/c)2 , where a = 20/7, b = 25/14, and c = 25/14.

It is evident that the numerical result converges to the exact value. The errors from the

Table 2.1: Areas computed from the coarea formula for bounded open sets
grid spacing
case
0.5
0.25
0.1
0.05 0.025 exact value
Sphere
4.00 4.00 4.15 4.17 4.18
4.189
Cylinder 22.50 23.25 24.49 24.84 25.01
25.133
Ellipsoid 37.75 37.97 38.10 38.17 38.16
38.163

cylinder are slightly larger than those from the sphere and ellipsoid because the cylinder has
non-smooth edges. However, the errors are small for all cases. Therefore, we conclude that
the mean value of the areas of the family of smooth solvent-solute surfaces indeed converges
to the area of the corresponding sharp surface. Thus, the present definition of the mean
surface area of an infinite family of smooth surfaces is an important generalization of the
classic concept of the area of a sharp surface.
48

2.3.2

Accuracy of the generalized PB solver

In this section, we investigate the accuracy of the proposed numerical solvers. The generalized geometric flow equation (A.8) has the same differential operator as the mean curvature
flow [20] except for the extra source terms. Previously, we have numerically proved that the
explicit Euler algorithm delivers the reliability and convergence of the solution of geometric
flow equations, and the finite central different scheme is of second order convergence in space
[17]. Here, we focus on the test of the accuracy of the generalized PB solver with the proposed dielectric function ǫ(S). Although the discretization form of the second-order finite
difference PB expression has been used for other continuous dielectric definitions [112], the
accuracy of this approach has not been examined. Moreover, it is worthwhile to validate the
generalized PB solver due to its different settings of dielectric function, i.e., the ǫ(S) profiles
generated by the geometric flow equation. For this purpose, we construct a benchmark test
of a simple one-ball system. We examine the convergence order and the accuracy of the
finite difference scheme in solving the generalized PB equation. In particular, we consider a
modified Poisson equation with a designed dielectric definition given by

ǫ(r) = ǫ1 Su(r) + ǫ2 · (1 − Su(r))
49

(2.27)

where r = (x, y, z), ǫ1 and ǫ2 are two constants to be determined, and


 1








b−
Su(r) =
−2 








 0

if
3

2 +y 2 +z 2
b−
x
 +3
b−a

x2 + y 2 + z 2 < a;

2
2 +y 2 +z 2
x
 if a ≤
b−a
if b <

x2 + y 2 + z 2 ≤ b;
x2 + y 2 + z 2 ;
(2.28)

where a < b. Note that through the definition of ǫ and Su(r), this designed case has the
same features of the dielectric definition as that in our model. The exact solution is designed
to be
φ0 (r) = cos(x) cos(y) cos(z).

(2.29)

Then the modified equation becomes

∇ · (ǫ∇φ) = ǫx ∇x φ0 + ǫy ∇y φ0 + ǫz ∇z φ0 + ǫ∇2 φ0
where
∇2 φ0 (x, y, z) = −3 cos(x) cos(y) cos(z)
∇x φ0 = − sin(x) cos(y) cos(z)
∇y φ0 = − sin(y) cos(x) cos(z)
and
∇z φ0 = − sin(z) cos(x) cos(y).

50

(2.30)

For ǫ, if a ≤

x2 + y 2 + z 2 ≤ b, we have


ǫi (r) = 6(ǫ2 − ǫ1 ) 



b−

x2 + y 2 + z 2

i

b−a


(b − a) x2 + y 2 + z 2




x2 + y 2 + z 2 − a
b−a




(2.31)



where i= x, y, z. Otherwise, ǫx = ǫy = ǫz = 0. Table 2.2 lists the computed errors under
different mesh sizes with a = 1 and b = 3. The standard absolute norm error measurement
L∞ is employed. Here ǫ1 is fixed to be 1 and ǫ2 is taken to be 80 or 10. The second order
convergence in space is observed for the scheme. Furthermore, it is found that a large ǫ2
may slightly deteriorate the convergence.
Table 2.2: Errors and convergence orders for the generalized PB solver (ǫ1 = 1)
ǫ2 = 80
ǫ2 = 10
spacing
L∞
order
L∞
order
1
0.22
0.13
0.5
8.13*10-2 1.65
3.18*10-2 2.02
0.25
2.06*10-2 1.99
7.97*10-3 2.00
0.125 5.44*10-3 1.94
1.98*10-3 2.01

2.3.3

Convergence of boundary profile and dielectric function

In the present model, the characteristic function S defines the solvent-solute boundary.
Consequently, it can significantly affect the solution of the generalized Poisson-Boltzmann
equation, the surface area and volume, and thus, plays a key role in the solvation free energy
calculation. To illustrate the evolution and the convergence of the generalized geometric
51

flow equation and corresponding S function, we solve Eq. (2.13) without the electrostatic
potential term in this test. However, the electrostatic solvation free energy at a given time can
att
still be calculated. The expression of attractive interaction Ui needs to be given explicitly
in order to solve the geometric flow equation. Here we consider the 6-12 Lennard-Jones (L-J)
att
pair potential to model Ui (see the description in Appendix A). All the calculations in this
work are carried out by using the WCA decomposition.
To illustrate the S profile and evolution, we consider a diatomic system with the van der
Waals radius 2.2˚ and coordinates (x, y, z) = (−3, 0, 0) and (1.4, 0, 0). The spacing and
A
time stepping are chosen as h = 0.25˚ and τ = h2 /4.5, respectively. The solvent probe
A
radius is set to 2˚, which is used for the initial value setting and constraint construction.
A
In fact a much small solvent probe radius can ensure the correct surface topology [20]. The
computational domain is set to [−8.70, 7.05] × [−5.7, 5.55] × [−5.7, 5.55]. Thus, the size of
computational system is 64×46×46. The L-J parameters are set as follows: σi is taken from
atomic radius and σs is chosen to be 0.65˚. We set well depth ǫi = 0.039 kcal/mol and bulk
A
density coefficient ρ0 /γ = 2, where, γ = 1/15kcal/(mol˚2 ). To compute the electrostatic
A
solvation free energy during the evolution of solvent-solute boundary, 1 unit charge is set on
the center of each atom. We choose the dielectric constants ǫm = 1 and ǫs = 80, respectively.
We set pressure coefficient p/γ = 0.2. A different γ is used for real systems.
The evolution process of diatomic solvent-solute boundary is depicted through a group of
cross section profiles of the S function in Figure 2.3 where the values of S from a set of points
of (x, y, 0.05) are described. The cross sections start with a relatively fat-shaped interface
which reflects the solvent accessible density. Here, S = 1 inside the molecular domain
and S = 0 in the solvent domain. Then the solvent-solute boundary is driven inward by
52

(T=0.07)

(T=1.0)

(T=1.5)

(T=5.0)

Figure 2.3: The evolutionary profiles of the S function at cross section (x, y, 0.05) in a
diatomic system plotted from four intermediate states.

53

intrinsic geometric curvature effect and external potential. At the same time, there appears
a transition region between the solvent and the solute. Finally a convergence is reached
with a balance among intrinsic geometric curvature effect, different potentials and enforced
constraints. To have a clear idea about the distribution feature of the S function, we draw a
cross line from the cross section graph at T = 5 along x = −0.75 which are shown in Figure
2.1, where the functions of S and 1 − S are described together. It can be seen that the S
function in the transition region is rather smooth. Once the S function is determined, the
dielectric function ǫ(S) is calculated according to Eq. (2.11). Here the dielectric function
ǫ(S) corresponding to the S function in Fig (2.1) is also exhibited in Figure 2.2. It has a
pattern similar to 1 − S but with different values. It is important to note that the dielectric
function ǫ(S) is also very smooth at the solute-solvent boundary. That is why the classical
finite difference scheme can be applied to solve the generalized PB equation without reducing
the accuracy of the solution.

Once the solution of the generalized PB equation is computed, the electrostatic solvation
free energy can be calculated. Therefore, the time history of the free energy along with the
evolution of solvent-solute boundary can be recorded. To illustrate the convergence pattern of
the solvation free energy, we compute the electrostatic solvation free energies in intermediate
states during the time evolution. The results are shown in Figure 2.4. In order to show
evolution histories of the surface area, volume and solvation free energy together in one plot,
we plot two linear functions F (Volume) and J(Area) which have the same time dependence
as the volume and the surface area, respectively. Here T denotes the time span and N = T
τ
represents the number of computational steps in the generalized geometric flow solver. It
is found that the solvation free energy decreases with respect to the time evolution, which
54

−50
−100
−150
−200
−250 0
10

Energy
F(Volume)
J(Area)
1

10

N

2

10

3

10

Figure 2.4: The time evolution histories of the electrostatic solvation free energy, F (Volume)
and J(Area), where F (Volume) = volume/5 − 180 and J(Area) = (surface area)/5 − 200.
is consistent with our theoretical formulation. It is observed that the solution of our model
converges to a steady state in terms of volume (˚3 ), area (˚2 ) and electrostatic solvation
A
A
free energy (kcal/mol). Moreover, to obtain the results at the steady state, N = 200 or
T = 3.5 is large enough to be taken as the stopping time in our geometry flow solver for
this system. Normally, it takes a longer evolution time for a large system to set down to the
steady state. The total integration time could be considerably shortened had a small probe
radius been used [20].

2.3.4

Consistency of iteration procedures

If the electrostatic potential effect is taken into account during the solvent-solute boundary evolution, the iteration procedure has to be used to update the electrostatic potential
repeatedly. As described earlier, there are two possible iterative approaches which can be
explored to solve the coupling system, in which the simple relaxation algorithm guarantees
55

the convergence of the algorithm. The question is whether these two approaches lead to the
same outcome.
Table 2.3: Comparison between two iteration approaches
2 atoms
Gly
Approach I Approach II
Approach I Approach II
Energy
-231.18
-231.18
-12.44
-12.44
Surface area
128.67
128.67
271.91
272.02
Volume
100.72
100.83
287.85
287.93

To study the consistency between these two approaches, the above mentioned diatomic
system is employed as well as a small molecule called glycerol triacetate (Gly) from a
set of 17 test compounds whose detailed information is given in Section2.4.

The self-

consistent iteration is performed until the electrostatic solvation free energy converges to
within 0.01 kcal/mol. The electrostatic solvation free energy(kcal/mol), surface area (˚2 )
A
and volume(˚3 ) resulted from these two different methods are compared. The results are
A
shown in Table 2.3. Here we take α = 0.5 for both methods. The electrostatic potential
φ is updated in every 15 steps of the generalized geometric flow integrations in Approach
II. It is evident that the results from these two approaches are almost the same. Therefore,
they can be alternatives for each other at least in small molecular systems. But for large
protein systems, as we mentioned, it is better to use the second approach to avoid the possible blowup in the generalized geometry flow caused by unpredictable large changes in the
temporary electrostatic potential. Thus, in the following tests and applications, the second
method is applied except specified.
In Approach II, the relaxation factor α and the number of intermittency Nstep need to
be determined. We are interested in knowing whether the relaxation factor α plays a role in
the final result. Similarly, it is important to know whether the Nstep makes a difference in
56

the converged result. We address these issues by examining the effects of α and Nstep on the
electrostatic solvation free energy, surface area and enclosed volume. The above mentioned
diatomic system is used here again.

Table 2.4: Effect of relaxation factor α on final results
α
0.1
0.2
0.5
0.7
0.8
Energy
-231.26 -231.18 -231.18 -231.18 divergence
Surface area 100.73 100.94 100.83 100.71
Volume
128.65 128.62 128.67 128.71

It is known that a stable α value is between 0 and 1 but can not be very close to 1. We
consider a number of α values in the diatomic system while keeping other settings fixed. Table
2.4 shows the electrostatic solvation free energy, surface area and volume for α = 0.1, 0.2, 0.5
and 0.7. It is found that the procedure diverges when α ≥ 0.8. However, convergence is
achieved as long as the relaxation factor α is small enough. Once the convergence is achieved
there is no much difference in the final outcome. We therefore take α = 0.5 in the following
tests and applications.
To study the effect of the number of intermittency, we take Nstep = 5, 10, 15, 40 and
100, while fixing α = 0.5 and other settings. The results are listed in Table 2.5. It is seen
that all values obtained from different number of intermittency are very close to each other.
However, a numerically too large or too small Nstep is not preferable. If Nstep is too large,
Approach II goes back to the first one. If Nstep is too small, the iterative process may stop
too early because the perturbation in each iteration is so weak that the convergence criteria
is satisfied unexpectedly sometimes. In addition, small step number makes the whole process
computationally expensive.
57

Table 2.5: Effect of the number of
Nstep
100
40
Energy
-231.18 -231.19
Surface area 100.83 100.83
Volume
128.63 128.63

2.3.5

intermittency in Approach II
15
10
5
-231.18 -231.17 -231.11
100.83 100.89 101.07
128.67 128.48 128.63

Efficiency of the accelerated iteration procedure

We study the efficiency of the accelerated self-consistent iteration in this section. At the
beginning we analyze the CPU time usage based on an original combination of methods:
Biconjugate Gradient (BiCG) method for the generalized PB solver and a widely used explicit
scheme for the generalized geometric flow (GGF) solver. In addition, as commonly used in
the linear system of the PB equation, we take 10−6 as the initial convergence criteria and set
the first guess of the electrostatic potential in each generalized PB run to be 0. The above
mentioned diatomic and Gly systems will be utilized through this efficiency test. Table
2.6 lists the breakup of time spending in the different parts of the self-consistent iteration
procedure. It is seen that for these two systems the major computation cost lies in the
routines of the generalized PB solver and the generalized geometric flow solver (more than
90%). Therefore, the total time will be dramatically reduced when efficient accelerations are
achieved in the generalized PB solver and generalized geometric flow solver. Note that all
of the computations are performed on a SGI Altix 350 workstation with a 1.4 GHz Itanium
processor and 4 GB memory.

Additionally, we explore the improvement made to the

generalized PB solver, the generalized geometric flow solver, and consequently to the total
time cost. First, we combine an appropriate preconditioner with the iterative solver of the
linear system. Additionally, we make use of the prior electrostatic potential as a first guess
for the next PB run. Moreover, we obtain the approximations through the relaxation of the
58

Table 2.6: CPU time analysis from original schemes
2 atoms
Gly
Time(s) %
Time(s) %
Total
23.95
58.4
GF
4.95
21
11.31
19
PB
18.25
76
45.24
77
Other
0.75
3
2.03
4

convergence criteria of the linear solver. Finally, we employ the ADI scheme to integrate the
generalized geometric flow equation.
First of all, we do the following improvement: Take the prior potential solution as the first
guess of each run of the linear solver, then replace BiCG scheme with a combination of the
preconditioner and the iterative solver (ILU/OM), while keeping other settings unchanged.
Table 2.7: Speedup from adjustment of initial guess and
h
size
BiCG
BiCG1
1
17× 12× 12 0.0557(252) 0.0322(152)
0.5
32× 23× 23
0.775(419)
0.467(248)
0.25 64× 46× 46 17.676(841) 10.410(490)
0.125 127× 92× 92 525.74(2771) 263.11(1371)

preconditioner in PB solver
ILU/OM
Speedup
0.0371(50)
1.50
0.420(82)
1.85
6.947 (166)
2.54
130.76 (410)
4.02

Table 2.7 gives the total computational costs of the generalized PB solver in the diatomic
system as well as the total iteration numbers which are inside the parenthesis. The third
column lists the time spending for original schemes, the fourth one makes use of prior potential as a first guess and the fifth one records the time spending from the usage of the
preconditioner and the new first guess setting. It is seen that the gain of speedup is related
to the size of system: The larger size is the system, the more acceleration is achieved. For a
127× 92× 92 system, combination of the above two implementations can obtain a speedup
up to a factor of 4, while a single adjustment does not give much impressive improvement.
59

It is also found that although the total iteration number reduces dramatically by adding
the preconditioner, the total computational cost is reduced with a much smaller factor. The
reason is that the PB solver with a preconditioner takes more time in each step.
Next, we study the impact of the convergence criteria to the electrostatic solvation free
energy of the diatomic system. Table 2.8 summarizes the calculated electrostatic solvation
free energies and total time cost of the PB solver under different convergence criteria. It
indicates that 10−4 is good enough to deliver accurate results. In fact, 10−2 is still fine but
10−1 is clearly unacceptable. In this study, we take 10−4 as the convergence criteria of the
linear system in the following calculations except specified. Because it is able to save much
time compared to 10−6 while at the same time maintains the accuracy to a satisfied level.
In practical application, one might use 10−2 . A further reduction in computational time is
possible if one sets the probe radius to rp = 0.25rvdW , where rvdW is the van der Waals
radius [20].
Table 2.8: Influence of convergence criteria on electrostatic solvation free energy and computational time for the diatomic system
Criteria
10−8
10−6
10−4
10−3
10−2
10−1
Energy (kcal/mol) -231.17 -231.17 -231.19 -231.28 -231.07 -239.80
PB Time (s)
10.44
6.95
4.17
3.09
2.05
0.90

Finally, we implement the ADI scheme in the generalized geometric flow solver. Thus
we can use a much larger time increment than that used in an explicit scheme without the
stability concern. For example, if grid spacing is h = 0.25, the time step size can be taken
as large as 0.2 for the ADI scheme to be a good balance between accuracy and efficiency,
while it has to be less than 0.02 in the explicit scheme. The acceleration of the generalized
geometric flow solver can be found in Table 2.9, which is obtained by applying all of the
60

speedup strategies we have discussed to the diatomic and Gly systems. This table shows all
the time spending for major routines in the iterative process before and after the acceleration.
It indicates that speedup in the PB solver can reach a factor of 4 or 5. However, the speedup
in the total time is not as impressive as in the PB solver. It is about a factor of 3 in the Gly
system and about a factor of 2 in the diatomic system. The reason is that the acceleration
in the generalized geometric flow equation through the ADI can not have the same speedup
factor as that of the PB solver. The electrostatic solvation free energies are also given in the
table for a comparison before and after the speedup. Little difference in energies is observed
due to varying schemes and the approximation.
Table 2.9: Comparison of CPU time (s) in the iteration procedures with and without accelerations
2 atoms
Gly
without
with
Speedup
without with Speedup
Total
23.95
8.87
2.70
58.40
16.28
3.59
GF
4.95
3.67
1.35
11.13
5.57
2.00
PB
18.25
4.45
4.10
45.24
8.71
5.19
Other
0.75
0.75
2.03
2.00
Energy -231.17 -231.18
-12.44 -12.44

2.3.6

Impact of potentials in the geometric flow equation

The potential source terms in the generalized geometric flow equation include pressure, longranged attractive dispersion interaction and electrostatic potential. The solution (S) of the
generalized geometric flow reflects a balance between the intrinsic geometric curvature effect
and several external potentials at the equilibrium. In this section, we illustrate the impact
of involved potentials to the characteristic function S, and consequently to the solvation free
energy. Although in our model there is no explicit surface definition to be demonstrated, the
61

impacts of these potentials can be reflected by volume, area and the electrostatic solvation
free energy. In particular, if the flow is driven inward by a potential, the volume should
become smaller, and an outward driving makes the volume larger. These are true at least
during the early stage of the solvent-solute boundary evolution. In fact, they are not true for
the system near the equilibrium. The present study is carried out through two proteins (PDB
ID 1ajj and 1fca) from protein data bank (PDB). Their detailed coordinates and parameters
are given in the application section of 22 proteins. Without any potential term, the mean
curvature flow equation is driven purely by intrinsic geometric curvature effect, which leads
to the minimal molecular surface (MMS) [20]. With the MMS as a reference, each time we
use one additional potential term in Eq. (2.13) to produce a new characteristic function
S which will be used in the PB solver to calculate the electrostatic potential. In other
words, starting with the MMS, we attain the different characteristic functions S with either
pressure, attractive non-polar potential, or electrostatic potential separately. Only when
electrostatic potential term is taken into account, is it needed to run the self-consistent
iteration process for the solution of the coupled system. Table 2.10 gives the calculated
volume and electrostatic solvation free energy under each potential term. We also calculate
the solvation free energy when all the potentials are turned on. It is seen that all the
potentials involved here drive the flow inward so that there are more solvent components
between or around two spherical solutes when just an individual potential is turned on. This
is consistent with the experimental observations [236] and our previous studies [17].

62

Table 2.10: Effects of potentials on the solvent-solute boundary
1ajj
1fca
Volume Energy
Volume Energy
MMS
6601.9 -975.6
9345.7 -1082.2
Pressure
6195.1 -1032.1
8866.6 -1123.2
Attractive
5533.8 -1139.3
8107.2 -1192.8
Electrostatic
6585.8 -1061.4
9329.3 -1112.4
Total potential 5381.6 -1165.6
7886.8 -1211.9

2.4

Applications

We consider two types of problems in this section. First, we apply our new approach to a set
of 17 small molecules. Then, some protein examples are studied. The Dirichlet boundary
condition is used for both the generalized Poisson-Boltzmann equation and the generalized
geometric flow equation as in our previous calculations [259, 252, 93, 20]

2.4.1

Set of 17 test molecules

We apply our optimized surface model (OSM) of solvation to compute the solvation free
energies of a set of 17 small compounds. This test set was studied by Nicholls et al [160]
using a number of approaches, including quantum mechanics, PB theory etc. An important
aspect about this test set is that experimental data are available. Therefore, solvation free
energies predicted from our new model can be compared with both experimental values and
other numerical results. Moreover, these compounds are considered as a challenging test
set for computational methods because the existence of polyfunctional or interacting polar
groups, which lead to strong solvent-solute interactions.
In our calculation, we set the dielectric constants ǫm = 1 and ǫs = 80. We use γ as
fitting parameter, and its initial value is set to γ = 1/15 kcal/(mol˚2 ) to compute other
A
63

Table 2.11: Comparison of free energies (kcal/mol) for 17 compounds
Compound
glycerol triacetate
benzyl bromide
benzyl chloride
m-bis(trifluoromethyl)benzene
N,N-dimethyl-p-methoxybenzamide
N,N-4-trimethylbenzamide
bis-2-chloroethyl ether
1,1-diacetoxyethane
1,1-diethoxyethane
1,4-dioxane
diethyl propanedioate
dimethoxymethane
ethylene glycol diacetate
1,2-diethoxyethane
diethyl sulfide
phenyl formate
imidazole

Gnp
2.27
1.40
1.35
2.22
1.96
1.86
1.45
1.65
1.52
1.01
1.82
1.03
1.59
1.55
1.22
1.37
0.82

∆Gp
∆G
Exptl Error
-12.44 -10.16 -8.84 -1.32
-4.89 -3.49 -2.38 -1.11
-5.02 -3.68 -1.93 -1.75
-3.22 -1.00
1.07
-2.07
-9.20 -7.24 -11.01 3.77
-7.67 -5.81 -9.76
3.95
-4.22 -2.77 -4.23
1.46
-8.24 -6.59 -4.97 -1.62
-4.40 -2.88 -3.28
0.40
-5.65 -4.64 -5.05
0.41
-7.85 -6.03 -6.00 -0.03
-4.52 -3.50 -2.93 -0.57
-8.43 -6.84 -6.34
0.50
-4.31 -2.76 -3.54
0.78
-2.39 -1.17 -1.43
0.26
-7.84 -6.48 -4.08 -2.40
-11.27 -10.45 -9.81 -0.64

γ-dependent parameters, see Eq. (2.13). We choose ρ0 /γ = 2 by comparing the bulk
density 0.033˚−3 and the possible γ value. For micro-molecular systems, pressure p can
A
be very small and sometimes is neglected in the calculation [48]. But here we still take it
into account and set p/γ to 0.2. Note that in the numerical simulation, all ratio parameters
here are treated as dimensionless. For L-J parameters, σs is chosen to be 0.65˚ as a
A
good fitting solvent radius and σi is the solute atomic radii [235]. Note that due to the
continuum representation of solvent in our model, the 6-12 Lennard Jones potential formula
(A.2) differs from the standard version — the distance used in our formula is no longer
the distance between the centers of solute atoms and the centers of solvent atoms but the
distance between a specific position in the solvent area and the centers of solute atoms.
This should make the setting of well depth ǫi different from the ones taken from AMBER
64

or OPLS force fields. However, the performance of the L-J potential should be similar, i.e,
the value of the L-J potential in the solvent caused by a solute atom only depends on the
distance from the center of the atom. It implies that the value of L-J potential caused by
a solute atom should be a constant on the van der Waals surface of the atom. In other
σi +σs 12
σi +σs 6
words, ǫi
−2
= Di if r is on the vdW surface of the atom. Here
|r−ri |
|r−ri |
the constant Di should have different values for various types of atoms. For simplicity we
use a uniform constant D to determine the value of ǫi given σs and σi . In the present
calculation, we pick 1.0 for D and the WCA expression is chosen as the attractive van der
Waals potential. We choose grid spacing h = 0.25˚ and time stepping τ = h2 /4.5. Here, γ
A
(kcal/(mol˚2 )) serves as a fitting parameter, and its final value is 0.0065 kcal/(mol˚2 ).
A
A

Structure and charge information of the 17 compounds are adopted from those of Nicholls
et al [160] and can be obtained from the supporting information of their paper. In particular,
charges are taken from the OpenEye-AM1-BCC v1 parameters [115]. Atomic coordinates
and radii are based on their new parametrization called ZAP-9 in which certain types of
radii are adjusted by them from Bondi radii to improve the agreement with experimental
free energy. With these structures and charges parameters, the root mean square error
(RMS) obtained in their paper is 1.71± 0.05 kcal/mol via the explicit solvent model. And
the smallest RMS error of their single - conformer Poisson-Boltzmann approach is 1.87±0.03
kcal/mol [160]. Such a large RMS error indicates the challenge of this test set. Usually,
different surface definitions in implicit solvent models should have their own optimal radii
set. In particular, a continuous dielectric definition based model is supposed to have radii set
with larger values than those of a discontinuous dielectric model. Otherwise, the calculated
free energy does not give a good fitting to experimental data [217]. This also occurs in the
65

present model. Therefore, we multiply the radii from ZAP-9 by a common factor 1.1. It turns
out this treatment leads to a good agreement with experimental data in terms of electrostatic
solvation free energies and total solvation free energies. The results are summarized in Table
2.11, which gives a comparison between calculated and experimental values of solvation free
energies of 17 compounds. RMS error of the present model is 1.76 kcal/mol which is similar
to that of Nicholls et. al, i.e., 1.87 kcal/mol. This RMS error is competitive to that of
the explicit solvent approach (1.71±0.05 kcal/mol) under the same charge and structure
parameters set [160]. This may be credited to the more satisfactory nonpolar terms and
the enforcement of the potential driven geometric flow. Here, as expected, major errors
are from the calculation of benzamides which are between 3.5 and 4.0 kcal/mol, see Figure
2.5. Without these benzamide compounds, the RMS error drops from 1.76 kcal/mol to 1.24
kcal/mol. This problem with benzamides is likely due to radius adjustment for the carbonyl
oxygens and tertiary nitrogens in ZAP 9 under the OpenEye-AM1-BCC v1 charges [160]. In
other words, these large errors from benzamides can not be avoided if both OpenEye-AM1BCC v1 charge and corresponding optimized ZAP 9 radii are used in PB approaches. Based
on these considerations, one possible approach for improvement is to create a new charge
set more appropriate for the PB approach with the same ZAP radii. It may be realized by
introducing quantum mechanical corrections to our model to take care of charge density.
However, this aspect will be investigated in next chapter 3.

2.4.2

Solvation free energy of proteins

Validation by using a set of 17 molecules has shown that proposed differential geometry
based solvation model works well for the energy prediction of small compounds. Since small
66

2

OSM/ Nicholls’

0

Exptl vs OSM
Exptl vs Nicholls’

−2
−4
−6
−8

−10
−12
−12

−10

−8

−6
−4
−2
Exptl (kcal/mol)

0

2

Figure 2.5: Correlation between experimental data and the present optimized surface model
(OSM)(also results from Nicholls’) in electrostatic solvation free energies of 17 compounds.

molecules are accessible to more accurate computational means, such as quantum mechanical
calculations, one of the main purposes of developing the present optimized surface model
(OSM) is to attack relatively large macromolecules. To this end, we consider a test set
of proteins employed by Mei et al [152]. For this set, the total number of residues ranges
from 21 to 275. The initial structures of all proteins are taken from the protein data bank
(PDB). The hydrogen atoms, which are typically missing from the X-ray data, are added
to the structures to obtain full all-atom models with optimized hydrogen bondings. Partial
charges at atomic sites and atomic van der Waals radii in angstroms are assigned from the
CHARMM27 force field [146]. All of these operations, i.e, the transformation from PDB
files to PQR files, can be easily done with a software PDB2PQR. Parameters of the present
calculation are set in the same way as those for 17 compounds except for Nstep = 2. Similar
to the treatment of the 17-compound set, the radii from the CHARMM force field need to
be multiplied by a common factor of 1.1. Our results are summarized in Table 2.12.
67

For a comparison, The results of Mei et al are listed in Table 2.12 as well. Their results
obtained from the molecular fractionation with conjugate caps and conductor-like polarizable
continuum model (MFCC-CPCM). This is an approximate quantum approach that divides
the macromolecule into fragments, such that the quantum calculations at HF/6-31 G level
and B3LYP/6-31 G level can be applied. The solvation effect is estimated vis the polarizable
continuum method with the classic molecular surface [152]. It is seen from the table that
there are relatively large deviations, up to 28%, between results obtained by the present OSM
and those of the MFCC-CPCM. These derivations might due to the different methodologies,
computational environments and structures. In fact, the results from two different quantum
basis sets have up to 10% deviation for protein Amyloid. Another deviation between results
of two quantum basis sets is about 5% for the protein BPTI.
Table 2.12: Comparison of electrostatic solvation free energies (kcal/mol) obtained from the
MFCC-CPCM, the present model (OSM) and MIBPB.

Protein
RP71955
Amyloid
Crambin
BPTI
Calbindin
Ubiquitin
Lysozyme
Subtilisin

PDBID No.of residues
1RPB
21
1AMC
28
1CBN
46
1BPI
58
1CDN
75
1UBQ
76
2BLX
129
1SBT
275

MIBPB-III
-184.68
-861.65
-303.80
-1301.9
-2188.96
-1170.61
-1913.40
-1896.5

△Gp(kcal/mol)
MFCC-CPCM [152]
-267.60
-886.01(-798.72)
-361.52
-1332.71(-1263.52)
-2259.62
-997.02
-1887.71
-2062.2

OSM
-192.23
-852.68
-304.84
-1281.19
-2195.42
-1148.81
-1898.07
-2001.4

As the largest deviation between the results from the proposed OSM and that of the
MFCC-CPCM is quite large, we consider another independent approach, the MIBPB [259,
252, 93], to evaluate the present method. A specific MIBPB code, the MIBPB-III which
has the treatment of geometric and charge singularities [93], is employed in our calculations.
68

0

OSM(kcal/mol)

−500
−1000
−1500
−2000
−2500
−2500 −2000 −1500 −1000 −500
MFCC−CPCM(kcal/mol)

0

Figure 2.6: Correlation between MFCC-CPCM [152] and the present optimized surface model
(OSM) in electrostatic solvation free energies of 8 proteins.

MIBPB-III has been intensively calibrated in the past and is the only known second accurate
method for solving the Poisson-Boltzmann equation with both molecular surfaces and partial
charges represented by the Dirac delta functions. To deliver such an accuracy, the MIBPB-III
has built in the MIB scheme [261, 254] and Dirichlet to Neumann mapping [93]. Similar to
the present approach, the structural data of MIBPB-III is also prepared with the PDB2PQR
software. As such, we can eliminate the difference due to the different treatment of initial
data. However, the MIPPB utilizes the classic PB equation and the molecular surface, while
the present method has a generalized PB equation, and an optimized smooth surface. It
is seen from Table 2.12 that solvation energy results from the present OSM and from the
MIBPB have an excellent agreement on most proteins except for Subtilisin. For this protein,
the difference of energies from two methods is about 5%.
69

0

OSM(kcal/mol)

−500
−1000
−1500
−2000
−2500
−3000
−3500
−3500 −3000 −2500 −2000 −1500 −1000 −500
MIBPB−III(kcal/mol)

0

Figure 2.7: Correlation between MIBPB-III and the present model (OSM) in electrostatic
solvation free energies of 22 proteins

2.4.3

Twenty two proteins

Encouraged by the good consistency of the proposed method and the MIBPB-III, we further
compare these approaches by a larger set of protein molecules — twenty two proteins that
have been frequently used in previous studies [83, 252, 93, 17]. The implementation of these
two methods is the same as that described in the last section.
Table 2.13 shows the results from the present continuous dielectric model, denoted as
“Radii1” in the table, and those by MIBPB-III. It turns out that electrostatic solvation
energies obtained via our minimization process are very close to those based on the MIBPBIII. This can also be seen through Figure 2.7 which shows that the results between them are
quite linearly correlated. The correlation coefficient is 0.999.
It is still interesting to understand how important it is to use a slightly enlarged radius
in smooth surface models [217]. To this end, we carry out the present calculations by using
70

Table 2.13: Electrostatic solvation free energies for 22 proteins
△Gp (kcal/mol)
PDB-ID No. of atoms MIBPB-III Radii1 Radii0
1ajj
519
-1137.2
-1178.5 -1362.6
1bbl
576
-986.8
-965.94 -1158.7
1bor
832
-853.7
-871.4 -1066.5
1fca
729
-1200.1
-1200.6 -1340.9
1frd
1478
-2852.2
-2844.8 -3173.4
1fxd
824
-3299.8
-3291.9 -3496.9
1hpt
858
-811.6
-808.2 -1039.1
1mbg
903
-1346.1
-1328.2 -1535.4
1neq
1187
-1730.1
-1713.9 -2049.3
1ptq
795
-873.1
-866.2 -1064.5
1r69
997
-1089.5
-1072.7 -1294.0
1sh1
702
-753.3
-771.8 -973.8
1svr
1435
-1711.2
-1704.6 -2073.7
1uxc
809
-1138.7
-1125.7 -1350.9
1vii
596
-901.5
-892.0 -1052.1
2erl
573
-948.8
-935.8 -1067.3
2pde
667
-820.9
-843.0 -1049.3
451c
1216
-1024.6
-1020.6 -1291.8
1a2s
1272
-1913.5
-1900.3 -2155.0
1a7m
2809
-2155.5
-2179.8 -2666.1
1a63
2065
-2373.5
-2380.5 -2912.0
1vjw
828
-1237.9
-1226.6 -1411.4

the original CHARMM22 van der Waals radii, denoted as “Radii0”. This result is also listed
in Table 2.13. It is seen that results from the original CHARMM22 van der Waals radii can
have over 20% deviations from those of “Radii1”. This helps to come to a conclusion that
for continuous dielectric models, it is necessary to enlarge atomic radii obtained from widely
used force fields. Otherwise, the results will be inconsistent with those of other analysis. This
is in agreement with the observation in the literature [217]. The necessity of using larger
radii is also shown clearly in Fig 2.8 by the differences of electrostatic solvation free energies
obtained from the MIBPB-III and the present calculations with original radii (Radii0) or
71

600
MIBPB−Radii1
MIBPB−Radii0
Kcal/mol

400
200
0
0

5

10
15
Protein

20

25

Figure 2.8: Differences between electrostatic solvation free energies obtained from the
MIBPB and the present model with original radii (Radii0) or enlarged radii (Radii1).
enlarged radii (Radii1).
Additionally, it is useful to demonstrate that the electrostatic potential function computed in the present OSM can be illustrated at arbitrary isosurface of the characteristic
function S. This is done by first computing a sharp surface at a given S value, then projecting the φ value on the isosurface of a given S value. Figure 2.9 shows three plots of the
electrostatic potentials at S = 0.25, 0.5 and 0.75. A comparison of these potentials indicates

(S=0.25)

(S=0.50)

(S=0.75)

Figure 2.9: Surface potential display of one protein (PDBID: 1frd) at different isosurfaces
72

the fast/slow electrostatic potential changing regions in the solvent-solute boundary. These
regions are also interactive regions in the protein-protein or protein-ligand interactions.
Finally, it remains an important issue to further improve the computational efficiency,
although systematical efforts have been made in this work to reduce CPU cost. Since the
coupled generalized PB and geometry flow equations are needed to evolve self-consistently to
reach the steady state, it takes more CPU time for the present method to calculate the total
free energy than some existing approaches that compute the polar and nonpolar energies
separately.

2.5

Chapter conclusions

This chapter presents a novel differential geometry based solvation model. A crucial concept in the present model is the characteristic function or the description function of solute
molecules which is one inside the solute domain and zero inside the solvent. Near the
solvent-solute boundary, the characteristic function gradually changes from one to zero over
a region of transition. The exact position and width of the transition region are determined
by a variational framework, which is formulated based on the total solvation free energy.
As a key ingredient of the present framework, the total energy encompasses coupled polar
and nonpolar contributions. The polar solvation free energy functional is described by the
electrostatic theory at equilibrium, while the nonpolar solvation free energy functional consists of surface energy, mechanical work and attractive solvent-solute interactions. Both the
polar and nonpolar solvation free energies are coupled through the characteristic function
S. In the present work, geometric measure theory is utilized to convert the Lagrangian
formulation of the surface into appropriate Eulerian formulation. By variation of the total
73

solvation free energy functional with respect to the characteristic function and electrostatic
potential, a generalized geometric flow equation for the electrostatic potential and a generalized Poisson-Boltzmann equation for the characteristic function are obtained. Unlike the
standard Poisson-Boltzmann equation, the generalized Poisson-Boltzmann admits a smooth
dielectric profile governed by the generalized geometric flow equation, which provides a physical description of the true solvent-solute dielectric boundary, according to the variational
principle. The generalized geometric flow equation balances the intrinsic geometric curvature effect and external potential due to mechanical work, solvent-solute interactions, and
the electrostatic potential. The solution of the generalized geometric flow equation and the
generalized Poisson-Boltzmann equation leads to quantities for the direct evaluation of the
solvation free energy.

74

Chapter 3
Quantum formulation
The objective of this chapter is to incorporate a quantum mechanical description of charge
density into our earlier differential geometry based solvation model, which is described in
Chapter 2. To this end, we hope to develop a more accurate and self-consistent multiscale
approach for the solvation analysis of both small and large molecules. The advantages
of the present quantum formulation of the differential geometry based multiscale solvation
models are follows. First, compared with our earlier solvation models, the present model
is able to provide more accurate descriptions of charge arrangement during the solvation
process and leads to more accurate prediction of solvation free energies. Additionally, the
present multiscale model reduces the dependence of our earlier solvation models on the
existing molecular mechanical force field parameters, which are typically parameterized for
certain class of (macro-) molecular systems and may not be appropriate for other class
of molecules. Therefore, the present model can be applied to a wider class of molecules.
Moreover, compared with other existing QM based solvation models [227, 237, 42], the
present model avoids the use of unphysical solvent-solute interfaces. The solvent-solute
75

boundary in the present model is described by the differential geometry theory of surfaces.
Finally, a systematical framework is established to incorporate polar energy, nonpolar energy
and quantum energy into a total energy functional. The optimization of the total energy
functional leads to coupled governing equations for a set of important state functions, such
as electrostatic potential, electronic density, and solvent-solute boundary profile. This set
of state functions gives rise to theoretical predictions of solvation free energy, electrostatic
profile and solvent-solute interface of the solvent-solute complex.

This chapter is organized as follows. Section 3.1 is devoted to the theoretical formulation of our differential geometry based quantum model of solvation. We provide a detailed
description of various solvation free energy functionals. Three governing equations, i.e., the
generalized Poisson-Boltzmann equation, the potential driven geometric flow (i.e., generalized Laplace-Beltrami) equation, and the Kohn-Sham equation are derived from the total
energy functional via the Euler-Lagrange variation. Numerical methods and algorithms are
presented in Section 3.2. This section offers detailed schemes for the solution of the abovementioned three governing equations. The dynamical coupling of these three equations is
achieved by an efficient iterative procedure. A formula for the solvation free energy estimation is also derived from the multiscale total energy functional. The present multiscale
model is validated by numerical tests using a number of molecules in Section 3.3. To establish a valid approach, we have examined consistency of the electron density with the Poisson
equation. The unit conversion between conventions used in our Poisson solver and that in
a DFT software is discussed. The results from the present multiscale mode is compared
with those in our previous methods and those in the literature. Applications to three sets of
molecules are given in Section 3.4. Some of these sets are computationally challenging. We
76

demonstrate that the present model performs well in the prediction of solvation free energies.
This chapter ends with a conclusion.

3.1

Theory and model

In this section, we first prescribe the polar and nonpolar free energy functionals based on
our differential geometry theory of the solvent-solute interface introduced in section 2.1.1 of
Chapter 2. We then give an expression for the quantum mechanical energy of electrons. In
the present work, the quantum mechanical energy of electrons is also treated as a part of
the multiscale total energy for the solvation system. Governing equations for the solvation
process are derived by the Euler-Lagrange variational principle.

3.1.1

Charge density based polar free energy functional

The solvation process involves both intermolecular and intramolecular interactions. Solvation analysis has been following certain convention, which may not be precisely consistent
with that in other fields. Typically, solvation interactions are classified into polar type and
non-polar type. The polar type of interactions is often modeled by the Poisson-Boltzmann
(PB) equation with appropriate point charges at atomic central positions. In the original
electromagnetic theory, the charge source of the electric potential is to be “free charges”.
However, in biophysics, due to the atomistic nature of the description, the point charges are
obtained by the fitting of the electron density distribution of either a charged molecule or
a charge-neutral molecule into its atomic centers. Such point charge information is often
stored in the database of popular software packages, such as CHARMM [146]. Therefore,
the polar interactions include both charge and polarization effects inside the molecule. Note
77

that the effect of the rearrangement of electron charges during the solvation process needs
to be computed twice, once before and once after the solvation. Polar interactions are also
called electrostatic interactions. However, not all electrostatic interactions are described by
the PB equation. Strictly, the electrostatic potential solved from the PB equation represents Coulombic type of interactions between charges. However, many other intermolecular
interactions, such as London dispersion interactions, Debye (induced dipole) interaction, iondipole interactions and dipole-dipole interactions are also electrostatics in origin, and are not
represented by the PB equation [121, 1, 5, 6, 194].

Sharp and Honig [199] and Gilson et al. [97] have given a formulation for the electrostatic
free energy functional. However, their formulation is based on a given static sharp solventsolute interface. In the present work, we follow our earlier definition of differential geometry
based electrostatic free energy functional in Chapter 2

1
S ρm φ − ǫm |∇φ|2
2
Ω


Nc

0 e−φQi /kB T − 1  dr
− 1 ǫs |∇φ|2 − k T
+ (1 − S)
ni
B

2
i=1

Gp =

(3.1)

where Qi is the charge of ith ionic species, Nc is the total number of ionic species, kB is
the Boltzmann constant, T is the temperature, and n0 is the bulk concentration of the ith
i
ionic species. Here, ǫs and ǫm are the permittivity, or dielectric constants of the solvent
and solute domains, respectively. The ǫ is unity in vacuum, but assumes different values in
different environments. In solvation analysis, ǫ is usually set to 1 or 2 in the solute domain
and to 80 in the solvent domain. In Eq. (3.1), ρtotal is the total charge density of the
78

molecule and is given by

ρtotal = qn(r) − qnn (r)
= qn(r) − q

I

(3.2)

ZI δ(r − RI ),

where q is the unit charge of an electron, n(r) is the electron density, nn (r) is the nucleus density, and ZI and RI are the atomic number and the position vector of nucleus I,
respectively.
In Eq. (3.1), the term associated with S is the electrostatic free energy of the solute and
that associated with (1 − S) is the electrostatic free energy of the solvent. In our model,
the surface function S will be determined by the total energy optimization. Additionally,
non-polar solvent-solute interactions are modeled in the same framework as that in Section
1.1.4.

3.1.2

Quantum mechanical energy functional

In the present multiscale model, we need to evaluate the total charge density ρtotal (r) by
quantum mechanical principles or ab initio approaches. However, the ab initio calculation of
the electronic structure of a macromolecule is intractable at present due to the large number
of degrees of freedom. A vast variety of theories and algorithms have been developed in the
literature to reduce the dimensionality of this many-body problem. One of the simplest ab
initio approaches is the Hartree - Fock (HF) method, which replaces instantaneous Coulombic
electron-electron repulsion interactions with a mean-field average. A variational procedure is
used to minimize the energy. An alternative of the HF method is the density functional theory
(DFT), which is originated from the Thomas-Fermi model. DFT represents the electronic
79

structure (principally the ground state) of a many-body system as a functional of a single
electron density. As usual in many-body electronic structure calculations, the nuclei of the
molecule of interest are treated by the Born-Oppenheimer approximation (i.e., as fixed) in
DFT to generate a static external potential in which the electrons are moving. The selfconsistent iterations are utilized to minimize the total energy of the system. Recently, DFT
has become one of the most popular and versatile methods available in computational physics,
computational chemistry and computational biology. In the present work, we incorporate
the DFT description of the electronic structure of the solute molecule into our differential
geometry based solvation model.

Despite the improvement in computer hardware and software for the quantum mechanical calculation, computational costs are still a major concern for the QM simulation of large
molecules of interest. Therefore, so-called order-N algorithms [98, 169], in which the computer time and memory scale linearly with the simulated system size, become increasingly
important. Though the plane wave basis set has advantages over local basis sets in terms
of avoiding basis-set superposition error as well as convergence concerns, it is difficult to be
used in the implementation of the O(N) method in DFT. As such, a localized basis set is
normally taken to develop fully self-consistent O(N) DFT algorithms. Along this line, a software package named SIESTA (Spanish Initiative for Electronic Simulations with Thousands
of Atoms) was developed [209, 3]. It is based on a flexible linear combination of atomic
orbitals (LCAO) basis set and essentially perfect O(N) scaling. Therefore, it allows very
fast simulations using minimal basis sets and very accurate calculations with complete multiplezeta and polarized bases [169, 168]. Moreover, the pseudopotential is used in SIESTA
to avoid the calculation of core electrons and to achieve the expansion of a smooth (pseudo80

) charge density on a uniform spatial grid domain, which further accelerates the speed of
quantum calculations.

3.1.2.1

Kinetic energy

Combining DFT with our differential geometry based solvation formulation, we define the
kinetic energy functional as
2
Gkin [n] =

S(r)
j

2m

|∇ψj (r)|2 dr

where m(r) is the position-dependent electron mass,

(3.3)

h
= 2π with h being the Planck con-

stant, and ψj (r) are the Kohn-Sham orbitals. Here, the total electron density n is given
by
n(r) =
i

|ψi |2 ,

(3.4)

where the summation is over all the Kohn-Sham orbitals. Note that orbitals {ψj } are subject
to the orthonormality constraint


 1

∗
Sψi (r)ψj (r)dr =

 0

i=j

(3.5)

i = j.

Obviously, Eq. (3.5) is an approximation which is valid as long as the boundary represented
by the characteristic function S is sufficiently far away from atomic centers of the solute
molecule. This is true in our model.
81

3.1.2.2

Potential energy

Without external potentials, the electrostatic potential energy of nuclei and electrons can
be represented by the Coulombic interactions among the electrons and nuclei. There are
three groups of electrostatic interactions: interactions between nuclei, interactions between
electrons and nuclei, and interactions between electrons. Because of the Born-Oppenheimer
approximation, interactions between nuclei do not directly have a impact on the structure
of electrons in DFT.
According to the Coulombic law, the repulsive interaction between electrons can be expressed as the Hartree term

Uee [n] =

q 2 n(r)n(r′ ) ′
dr ,
ǫ(r)|r − r′ |

1
2

(3.6)

where q is again the unit charge of an electron, ǫ(r) is the position dependent electric permittivity, and r and r′ are positions of two interacting electrons. Equation (3.6) gives rise
to a nonlinear function in terms of electron density n. Therefore, the problem of solving the
electronic structure has to be resolved by self-consistent iterations.
Additionally, the attractive interactions between electrons and nuclei are given by

Uen [n] = −

I

q 2 n(r)ZI
.
ǫ(r)|r − RI |

(3.7)

Finally, we write the total potential energy functional as

Gpotential =

Ω

S(r) Uee [n] + Une [n] + EXC [n] dr,
82

(3.8)

where the last term, EXC is the exchange-correlation potential, which includes all the manyparticle interactions in the solute molecule. In general, the exact form of the exchangecorrelation potential is not known. There are good approximations in the practical applications, such as the local-density approximation, the local spin-density approximation,
and generalized gradient approximations. A detailed elaboration of the exchange-correlation
potential is beyond the scope of the present work.

3.1.3

Total free energy functional

Intuitively, it may appear that the total free energy functional is the summation of the
polar, non-polar, kinetic and potential energy. However, such an approach will lead to some
double counting because of the coupling among different energy terms. For example, the
electrostatic energy depends on the charge density, which, in turn, depends on the kinetic
and potential energies of electrons. Additionally, the electrostatic potential serves as an
unknown in the polar energy functional, meanwhile it serves as an input in the potential
energy of electrons. To see this connection, we need to solve the Poisson equation in vacuum
(ǫ = 1)
−∇2 φv (r) = ρv
total (r),

(3.9)

where φv is the electrostatic potential in vacuum and ρv
total = nv + nn with nv (r) being
the electron density in vacuum. The solution of Eq. (3.9) is

φv (r) =

qnv (r′ ) ′
dr −
|r − r′ |
83

I

qZI
.
|r − RI |

(3.10)

Note that Eq. (3.10) is the exact total Coulombic potential of electron-electron interactions
and electron-nucleus interactions. As such, we do not need to include Uee [n] and Uen [n]
terms in the total free energy functional.
Finally, we propose a multiscale total free energy functional for biomolecules at equilibrium

Gtotal [S, φ, n] =
{γ|∇S(r)| + pS(r) + (1 − S(r))ρ0 Uss
Ω


Nc
1
n0 e−Qi φ/kB T − 1 
+(1 − S(r)) − ǫs |∇φ|2 − kB T
i
2
i=1



2
1
2 + S(r) 
2 + E [n] dr
+S(r) ρm φ − ǫm |∇φ|
|∇ψj |
XC

2
2m
j

(3.11)

where the first row is the non-polar energy functional, the second row is the electrostatic
energy functional and the last row is the electronic energy functional. As discussed above, the
term ρtotal φ also contributes to the Coulombic potentials of electron-electron and electronnucleus interactions. This total free energy functional provides a starting point for the
derivation of governing equations and a basis for the evaluation of solvation free energies.

3.1.4

Governing equations

The total free energy functional (3.11) is a function of characteristic function S, electrostatic
potential φ and electron density n. The governing equations of these quantities can be obtained by the first variation of the total free energy functional (3.11). From the mathematical
point of view, there should exist optimal functions S(r), φ(r) and a set of orbitals {ψj } at the
equilibrium state in which the solvation free energy is optimized. The variational procedure
84

for S(r), φ(r) and {ψj } is described below.
First, by the variation of Eq. (3.11) with respect to the electrostatic potential φ, we have
δGtotal
= 0⇒
δφ
Nc
Sρm + ∇ · ([(1 − S)ǫs + Sǫm ]∇φ) + (1 − S)

i=1

n0 Qi e−Qi φ/kB T = 0
i
(3.12)

The Euler-Lagrange equation is used in the above variation. Equation (3.12) is the generalized Poisson-Boltzmann (GPB) equation [243, 46]
Nc
−∇ · (ǫ(S)∇φ) = Sρtotal + (1 − S)

i=1

n0 Qi e−Qi φ/kB T ,
i

(3.13)

where the dielectric function is given by

ǫ(S) = (1 − S)ǫs + Sǫm .

(3.14)

This is a smooth function. It is clear that the GPB equation utilizes a smooth dielectric
profile. There is a smooth transition region for the dielectric function to change from ǫs to ǫm .
Therefore, the solution procedure of the present GPB equation (3.13) avoids many numerical
difficulties of solving elliptic equations with discontinuous coefficients [258, 261, 260, 254, 253]
in the classical PB equation. Furthermore, in a solvent without salt, the GPB equation is
simplified to the generalized Poisson equation

−∇ · (ǫ(S)∇φ) = Sρtotal .
85

(3.15)

Both Eqs. (3.13) and (3.15) are similar to our earlier results in Chapter 2. However, in
the present multiscale model, the charge source ρtotal is to be determined by solving the
Kohn-Sham equations, rather than by the fixed charges ρfix =

j qj δ(r − rj ).

Additionally, by the variation of Eq. (3.11) with respect to the surface function S, we
have
δGtotal
=0⇒
δS
∇S
1
−∇ · γ
+ p − ρ0 Uss − ǫm |∇φ|2 +
|∇S|
2
Nc
+kB T
n0 e−Qi φ/kB T − 1 + ρm φ +
i
i=1

1
ǫs |∇φ|2
2
2
j

2m

|∇ψj |2 + EXC [n] = 0
(3.16)

In Eq (3.16), ∇ ·

γ ∇S
|∇S|

is a generalized Laplace-Beltrami operator, which is a gener-

alization of the usual Laplacian operator to a smooth manifold of macromolecular surface
[17, 243]. In general, γ can be a function of the position γ = γ(r) to account for the surface
hydrophobicity at different locations of the macromolecule. For simplicity, it is treated as a
constant in our present computation. By solving Eq. (3.16), we obtain a “physical solventsolute boundary” (S). As discussed in earlier work [20, 17, 243], the solution of this elliptic
partial differential equation (PDE) can be attained via a parabolic PDE

∂S
∇S
= |∇S| ∇ · γ
∂t
|∇S|
86

+V ,

(3.17)

where the generalized “potential” V is defined as

V

1
1
= −p + ρ0 Uss + ǫm |∇φ|2 − ǫs |∇φ|2 − kB T
2
2
2
−ρtotal φ −

j

2m

Nc
i=1

n0 e−Qi φ/kB T − 1
i

|∇ψj |2 − EXC [n]

(3.18)

where the electronic potentials in last row do not contribute much to V at equilibrium.
This is due to the fact that they are essentially confined inside the solute molecular domain.
Note that Eq. (3.17) has the same structure as the potential driven geometric flow equation
defined in Chapter 2. As t → ∞, the initial profile of S evolutes into a steady state solution,
which solves the original Eq. (3.16) with an optimal surface function S.
Finally, to derive the equation for the electronic wavefunctions, we need to incorporate the
constraint as shown in Eq. (3.5) into the total energy functional. This can be easily done
with a family of Lagrange multipliers

i Ei δij −

∗
Sψi (r)ψj (r)dr . Therefore, by the

∗
variation of Eq. (3.11) with respect to the wavefunction ψj (r) and subject to the constraint,
we have
δ Gtotal +
2
−

2m

i Ei δij −
∗
δψj

∇2 + Ueff

∗
Sψi (r)ψj (r)dr

ψj = Ej ψj ,

=0⇒
(3.19)

where the Lagrange multiplier constants Ei have been interpreted as energy expectation values. Equation (3.19) is the Kohn-Sham equation in which the effective Kohn-Sham potential
is defined as
Ueff (r) = qφ + VXC [n],
87

(3.20)

where VXC [n] =

dEXC [n]
with qφ being the potential contribution from Coulombic interdn

actions. It is to be calculated by the GPB equation (3.12) with a given total charge density.
Apparently, Eq. (3.19) does not directly depend on the surface function S, so that existing
DFT packages can be used in our computations with a minor modification as described in
Section 3.2.2.
It is seen that the generalized Poisson-Boltzmann equation (3.13), the generalized LaplaceBeltrami equation (3.17) and the Kohn-Sham equation (3.19) are strongly coupled to each
other. Therefore, these three equations have to be solved by appropriate iterative procedures.
This aspect is discussed in the next section.

3.2
3.2.1

Numerical methods and algorithms
Solution of the generalized Poisson-Boltzmann equation

The solution of generalized Laplace-Beltrami equations has been studied in appendix A,
including the details of some discretization schemes.
In solvation analysis, the generalized PB equation (3.15) is subject to the Dirichlet boundary condition [93]
Na
φ(r) =
j

qj
,
ǫs |r − rj |

∀r ∈ ∂Ω,

(3.21)

where qj is the total fixed charge of the jth solute atom. One option is to use the point
charges from a force field model such as CHARMM. However, in the present work, we
88

consider the following Dirichlet boundary condition

φ(r) =

ρtotal (r′ ) ′
dr ,
ǫs |r − r′ |

∀r ∈ ∂Ω,

(3.22)

where the boundary condition is nonlinear — it depends on the electron density n and thus
needs to be implemented iteratively.
The standard second order center difference scheme is applied in this study to solve Eq.
(3.15). An accurate solution can be obtained due to the continuous dielectric definition ǫ(S).
Let the pixel (i, j, k) represent the position (xi , yj , zk ). The discretized form of Eq. (3.15)
is

1
ǫ xi + hx , yj , zk [φ(i + 1, j, k) − φ(i, j, k)]
2
1
1
+ ǫ xi − hx , yj , zk [φ(i − 1, j, k) − φ(i, j, k)]
2
h2
x
1
+ ǫ xi , yj + hy , zk [φ(i, j + 1, k) − φ(i, j, k)]
2
1
1
+ ǫ xi , yj − hy , zk [φ(i, j − 1, k) − φ(i, j, k)]
2
h2
y
1
+ ǫ xi , yj , zk + hz [φ(i, j, k + 1) − φ(i, j, k)]
2
1
+ ǫ xi , yj , zk − hz [φ(i, j, k − 1) − φ(i, j, k)]
2

1
h2
z

= −S(i, j, k)ρm (i, j, k)
(3.23)

where hx , hy and hz are the grid spacings at x, y and z directions, respectively. Here,
ρtotal (i, j, k) is the charge density at grid point (xi , yj , zk ), which is calculated from the
electronic charge density n(r) and nucleus density nn . The implementation of ρtotal will
89

be discussed in the next paragraph. As such, the discretized PB equation can be converted
into the standard linear algebraic equation system of the form AX = B, where X is the
solution of interest, A is the discretization matrix and B is the source term associated with
the charge density. It has been shown previously that the PB solver is able to deliver the
designed second-order accuracy in Chapter 2.

On the right hand side of Eq (3.23), the charge density at each grid point should be
given. As an efficient approach, atomic charges have been widely used to approximate the
charge density of electrons and nuclei, especially for large molecules of general interest.
Therefore, the point charge approach has gained much popularity in PB solvers as well as
PB applications [147, 12, 112, 116]. Nevertheless, charge assignment at atomic centers is a
nontrivial issue. The deficiencies of the atomic point charge approach have been discussed
in the Introduction. The direct implementation of the quantum mechanical charge density
can avoid errors caused by the atomic point charge approximation. Moreover, this approach
frees us from the electrostatic potential fitting process. To carry it out in the finite different
scheme, the total charge density ρtotal (i, j, k), which consists of the electron density n(r) and
nucleus density nn , needs to be prescribed at each grid point of the computational domain.
In particular, the nucleus density nn (r), which is considered as stationary and located at
the center of atoms, can be distributed to the nearest eight neighboring grid points by the
second order interpolation (i.e., the trilinear mapping). The distributed nucleus core point
charges are converted into the nucleus charge density by dividing point charges with the
volume of the unit grid. Finally, the total charge density at each grid point is obtained by
the summation of nucleus density and the electronic charge density which is directly taken
from the quantum mechanical calculation.
90

However, a new issue arises from the above treatment of nuclei. Since each nucleus core
charge is split into its eight neighboring grid points, it is easy to find out that short range
interactions are biased and self-interaction energies are artificially introduced. This is due
to the interactions of grid charges within one single atom. It exists even in the absence of
solvent and any other charges. Apparently, this is a pure artifact due to the finite difference
approach and must be eliminated. Within the partial charge approach the artifact can be
canceled out mainly by calculating the PB equation twice, one in vacuum and the other in
the solvent. It turns out that this strategy also works well here. Numerical tests regarding this cancellation of self-interaction energies are demonstrated later. It is important to
point out that numerically if one implements the quantum mechanical calculation with a
non-frozen core method, the remaining error from the self-interaction cancellation is still too
large to be neglected. In other words, the above cancellation strategy may fail when one
applies a non-frozen core approach. Therefore, frozen core approaches, such as pseudopotential methods, must be applied in our quantum calculations here. Because frozen core
approaches dramatically reduce the number of charges in each nucleus and thus implicitly
decease implementation errors.

The biconjugate gradient method is a good choice in solving the PB equation. However,
as we have demonstrated in Chapter 2, the combination of stabilized biconjugate gradient
method (BiCG) and the blocked Jacobi preconditioner (BJAC) from PETSc (http://www.mc
s.anl.gov/petsc/petsc-as/), as well as the combination of the orthomin method (OM) and
the incomplete LU factorization preconditioner (ILU) from SLATEC, speeds up the process
of the PB solution. In this study, we apply the combination of ILU and OM from SLATEC.
In our iteration procedure, the prior electrostatic potential is taken as a good initial guess
91

for the followed linear system solving procedure. It turns out that this treatment makes the
generalized PB solver converge much faster than simply setting the initial guess to be 0 [46].
Additionally, the convergence tolerance is set to be 10−4 as a good compromise between the
accuracy and efficiency.

3.2.2

Solution of the generalized Kohn-Sham equation

The generalized Kohn-Sham equation (3.19) admits all-electron and all-nucleus potentials.
The direct solution of Eq. (3.19) is very expensive for macromolecules. Therefore, further
simplifications are necessary. In particular, because classical DFT methods have been developed in the past few decades, the solution of Eq. (3.19) needs to make use of existing DFT
methods.
Note that the Coulombic potential functionals shown in Eqs. (3.6) and (3.7) involve
spatially varying dielectric constants, which reflect the solvation process. The related spatially varying electrostatic potential is built in the generalized Poisson-Boltzmann equation
(3.13), whose solution gives rise to the electrostatic potential energy qφ used in the generalized Kohn-Sham equation (3.19). In contrast, the standard Kohn-Sham equation is for a
molecular system in vacuum and its Coulombic potentials are of the form of qφv where φv
is given by Eq. (3.10) with the total charge density in vacuum described in the next section.
The effective potential in the generalized Kohn-Sham equation (3.19) can be written as

0
Ueff [n] = qφ + VXC [n] = qφRF + Ueff (r),

(3.24)

φRF = φ − φ0

(3.25)

where

92

is called the reaction field potential. Here φ0 is the solution of the Poisson equation in the
homogeneous medium with the charge source ρtotal (r)

−∇ · ǫ0 ∇φ0 (r) = ρtotal (r),

(3.26)

where ρtotal (r) is obtained from the generalized Kohn-Sham equation (3.19). In Eq. (3.24),
0
Ueff (r) is the Kohn-Sham potential
0
Ueff (r) = qφ0 + VXC [n].

(3.27)

0
In the present work, Ueff (r) is represented by the traditional Kohn-Sham potential. Consequently, a vast variety of computational techniques developed for the traditional Kohn-Sham
DFT can be utilized in the present work. What we need to do in solving the generalized
Kohn-Sham equation (3.19) is to add a reaction field potential qφRF to an existing KohnSham DFT solver.

The most important issues in the solution of the Kohn-Sham equation are the selection of
the exchange-correlation potential and the use of the pseudopotential. The pseudopotential
approach eliminates the complicated effects of core electrons and allows the expansion of a
smooth (pseudo-) charge density on a uniform spatial grid. In this approach, the chemically
active valence electrons are dealt with explicitly, while the core electrons are ‘frozen’ and
considered together with the nuclei as fixed non-polarizable ion cores. With the pseudopotential approximation, the formalism of the total energy functional needs to be modified,
93

which leads to the following expression of a Kohn-Sham effective potential [209]

0
Ueff (r) =

I

local (r) +
VI

I

nonlocal + V H (r) + V (r)
VI
XC

(3.28)

where V H (r) and VXC (r) are total Hartree and exchange-correction potentials, respectively.
local and V nonlocal are the local part and the nonlocal part of the pseudopotential
Here, VI
I
of atom I. For elaborated discussions of the above potentials, we refer the reader to an
excellent review [209].
In the present work, SIESTA (Spanish initiative for the electronic structure of thousands
of atoms), a quantum mechanical package of high efficiency, is utilized for solving our generalized Kohn-Sham equation (3.19). SIESTA possesses the ability to perform the density
functional theory (DFT) simulations of more than a thousand atoms. The details of the
package has been extensively described [209]. It develops a self-consistent density functional
method using the standard norm-conserving pseudopotential and a flexible numerical linear combination of atomic orbital (LCAO) basis sets with essential perfect O(N) scaling,
in which the computer CPU time and memory scale linearly with the simulated system
size. The exchange and correlation are treated within the Kohn-Sham DFT. Both the local
density approximation and local spin density approximation (LDA/LSDA), as well as the
generalized gradient approximation (GGA) are allowed. Moreover, SIESTA permits very
fast simulations by using minimal basis sets as very accurate calculations with complete
multiple-zeta and polarizable bases. Therefore, it can provide a general scheme to perform
quantum calculations with requirements ranging from being very fast to being very accurate.
For all of the simulations in the present work, the default double-ζ plus single polarization
(DZP) bases are used. The MeshCutoff is set as 125 Rydberg and the LDA is applied. The
94

SolutionMethod is set to be ‘diagon’.

3.2.3

Evaluation of the solvation free energy

The solvation free energy is the energy required or released from the transfer of a unit of
solute molecules from vacuum to a solvent. By definition, it is calculated by the difference
of the total energies in solution and in vacuum

△Gtotal = Gtotal [S, φ, n] − Gvacuum [φv , nv ]

(3.29)

where φv is the electrostatic potential in vacuum and nv is the solute electronic density in
vacuum, which is defined in terms of the electronic wavefunctions of the solute in vacuum
v
ψj (r)
v
|ψj (r)|2 .

nv (r) =
j

(3.30)

In Eq. (3.29), Gtotal [S, φ, n] is given in Eq. (3.11) and Gvacuum [φv , nv ] denotes the total
energy functional in vacuum

1
2
ρv
total φv − 2 ǫ|∇φv |

Gvacuum [φv , nv ] =

2

+
j

(3.31)


v
|∇ψj |2 + EXC [nv ] dr
2m

where ρv
total = qnv − qnn is the total charge density in vacuum. For simplicity, we have
omitted the ionic density kB T Nc n0 (e−Qi φv /kB T − 1) in Eq. (3.29). Note that the
i=1 i
variation of Gvacuum [φv , nv ] gives rise to the standard Poisson equation (3.9) and the
95

standard Kohn-Sham equation
2
−

2m

v
∇2 + Ueff

v
v v
ψj = Ej ψj ,

(3.32)

v
v
where Ej and ψj are appropriate eigenvalue and eigenfunction of Hamiltonian H v =
2
v
− 2m ∇2 + Ueff .
However, there is a technical difficulty in the direct evaluation of Gtotal [S, φ, n]. Namely,
the integration of the quantum mechanical terms in Eq. (3.11) requires the S function profile,
which is not involved in most Kohn-Sham DFT software packages. Therefore, in the present
work, we evaluation the solvation free energy by the following approximation

△Gtotal = Gnp + ∆Gp + ∆GQM

(3.33)

where Gnp , ∆Gp and ∆GQM are the non-polar, polar and quantum mechanical contributions, respectively. The non-polar solvation free energy does not exist in vacuum, and its
form in solution is given by

Gnp [S] =

[γ|∇S(r)| + pS(r) + ρ0 (1 − S(r))Uss ] dr.

(3.34)

By using the Gauss’ divergent theorem and integration by parts, it is easy to show that the
polar solvation energy is given by

Gp [S, φ, n] =

1
ρ
φdr.
2 Ωs total

Similarly, the polar solvation energy in vacuum is 1
2
96

(3.35)

ρv
total φv dr. Therefore, one may com-

pute the change of the polar solvation energy by 1
2

ρv
total φv dr . How-

Ωs ρtotal φdr −

ever, this expression leads to a situation that the quantum mechanical contribution ∆GQM
cannot be evaluated in SIESTA because of the lack of required potential terms. Additionally,
such an expression is inconsistent with the conventional electrostatic solvation free energy of
the form

∆Gp =

1
1
ρtotal [φ − φ0 ] dr =
ρ
φ dr.
2 Ωs
2 Ωs total RF

(3.36)

Therefore, in the present solvation analysis ∆Gp is calculated by Eq. (3.36), which leads
1
to two remaining electrostatic potential terms 2

Ωs ρtotal φ0 dr −

ρv
total φv dr . These

terms are combined with the rest of the quantum energy functionals to compute the change
of the quantum mechanical energy as

∆GQM =
j

v
v
< ψj |H 0 |ψj > − < ψj |H v |ψj > .

(3.37)

2
0
where H 0 = − 2m ∇2 + Ueff . Note that wavefunctions {ψj } are computed with the full
Hamiltonian in the solution. The main advantage of the quantum mechanical energy change
given in Eq. (3.37) is that it can be easily computed by using existing DFT software packages
as discussed in the last section.
The current formula of the solvation free energy is systematically derived from the framework of the differential geometry based solvation model. It consists of the non-polar energy
Gnp , the electrostatic solvation free energy ∆Gp , and the change of the solute self-energy
∆GQM due to the redistribution of electrons in the solvation process. It is of interest to see
that the formulation of the present solvation analysis is consistent with that in the litera97

ture [224, 237, 238], which is basically computed by using a good chemical intuition. The
reliability and accuracy of the current model are further validated by a comparison of the
present prediction with experimental data, as well as with that in the literature in Section
3.3.

3.2.4

Dynamical coupling of involved PDE equations

As described in Section 3.1.4, the total charge density in the solution is obtained by solving
the Kohn-Sham equation in the presence of the reaction field potential φRF = φ − φ0 ,
which is computed by solving the PB equation and the Poisson equation, i.e., Eqs. (3.13)
and (3.26). On the other hand, the solution of the PB equation requires the quantum
mechanically calculated charge density, the surface profile S, and the dielectric profile ǫ(S)
which are generated by solving the generalized Laplace-Beltrami equation (LBE). Moreover,
the potential in the generalized Laplace-Beltrami equation contains the terms associated with
the electrostatic potential from the PB equation and the charge density from the Kohn-Sham
equations. In principle, the Laplace-Beltrami equation, the generalized PB and Kohn-Sham
equations need to be solved simultaneously until the convergence is reached, i.e, the solvation
energy of two runs agrees with each other within a prescribed tolerance. This can be achieved
via a self-consistent iteration procedure.
In practice, we adopt an inner-outer iterative strategy to implement the self-consistent
procedure. The inner iterations concern the solution of the coupled generalized PB equation
and the Laplace-Beltrami equation. These iterations have been carried out in Chapter 2,
except for the different representation of the continuous charge density. In the present work,
the inner iterations are combined with the solution of the Kohn-Sham equation during the
98

Initiate ρtotal in SIESTA
Coupled PB and LBE for φ
Solve for φRF
Update ρtotal in SIESTA
|∆Gnew − ∆Gpre | <Tolerence?
total
total
yes

Begin new loop
no

Stop and output Gtotal
Figure 3.1: Flowchart of the numerical solution of the coupled PDEs.
outer iterations. More details are described in the following and can be seen from a work
flow chart in Figure 3.1 as well.
Step 0 (Generation of the solute quantum energy in vacuum and initialization of the
charge density): We carry out a quantum mechanical calculation in vacuum with SIESTA
to obtain an initial total charge density. The solute quantum energy in vacuum is recorded
for computing ∆GQM in the output step.
Step 1 (Inner iteration of the coupled PB and Laplace-Beltrami equations): Given a total
charge density distribution, as described in Chapter 2, a temporary electrostatic potential φ
can be generated by solving the generalized PB equation with a temporary S. It is followed
by the evolution of time-dependent generalized Laplace-Beltrami equation for a number of
time iterations. With the updated intermediate S, one can update the electrostatic potential
via the PB equation. This cycle repeats until the electrostatic solvation energy converges
99

within a pre-determined criterion. Note that the relaxation algorithm should be used to
guarantee the convergence.
Step 2 (Generation of a reaction field potential): Solve the PB equation in a homogeneous
medium with the same total charge density as that in the previous step. Then the reaction
field potential φRF is obtained by the difference between the electrostatic potential from the
previous step and the current Poisson calculation in the homogeneous medium.
Step 3 (Solution of the Kohn-Sham equation): Run the SIESTA program again to obtain
a new total charge density by incorporating the computed reaction field potential into the
Kohn-Sham Hamiltonian.
Step 4 (Calculation of the solvation energy and convergence check): Calculate the total
solvation free energy with the resulting φ, S and {ψj }. Go back to Step 1 until it converges.
When convergence is reached, one ends the iteration and outputs the solvation free energy.
To output the charge density in SIESTA, one needs to set SaveRho to be true in the
input fdf file, while SpinPolarized is false according to the fact that all tested molecules
in this work are neutral. Eventually, SIESTA generates an XV file to store lattice vectors
and atomic positions, together with an RHO file to record the values of the charge quantity
on the grid points. However, the standard input for our PB solver is a Gaussian CUBE
format file which contains the origin, grid spacing, atomic coordinates and charge densities.
Therefore, a CUBE formate file must be created based on the information from the XV file
and RHO file to transfer the charge density data from SIESTA to the PB solver. It can be
carried out by a subroutine grid2cube.x, which is included in the SIESTA package. During
the translation, the coordinates of the origin are shifted to make the molecule roughly appear
at the center of the computational domain.
100

After the reaction field potentials are obtained by the PB solver, they are regarded as
an external solvent potential effect and must be incorporated into the Kohn-Sham quantum
calculation in SIESTA. As such, reaction field potentials have to be passed into SIESTA
during the self-consistent iteration process. It can be done by adding them into the variable of
total potential named Vscf in the subroutine file dhscf.F under “Siesta-3.0-b/Src”. Therefore,
a data translation procedure is required to pass the reaction field potential from the PB solver
into SIESTA. Furthermore, attention needs to be paid to the unit conversion. In particular,
the unit of distance used in SIESTA is Bohr, while it is Angstrom in the PB solver. The
units of potential are ec /Angstrom and Rydberg/ec in the present PB solver and in SIESTA,
respectively. Here ec denotes the fundamental charge used as the unit of a point charge in
both the PB solver and SIESTA.

3.3

Numerical test and validation

This section provides systematical validations for the proposed model and computational
algorithms. The performance of SIESTA has been tested and described in the literature
[209]. The generalized Laplace-Beltrami equation (3.17) has the same differential operator
as our earlier mean curvature flow [20] except for the extra source terms. Previously, we have
numerically proved that the explicit Euler algorithm delivers the reliability and convergence
of the solution of the Laplace-Beltrami equation, and the finite central difference scheme is
of second-order convergence in space [17]. For the PB solver, given the partial charges, it
has also been numerically proved to be of second-order convergence in Chapter 2. However,
here the source term in the PB equation is no longer represented by the partial charges
adopted from existing molecular mechanical force fields, such as AMBER or CHARMM. In
101

this work, the source term is represented by the charge density obtained directly from the
quantum calculation. There are new concerns from this different charge strategy. First of
all, regarding the distribution of nucleus charges, which are much larger than partial charges,
it is crucial to know whether the self-interaction energy artifact within a single atom leads
to non-neglected bias even after the treatment of the energy cancellation. Secondly, it is
important to check whether the implementation is correct in terms of the data translation
and unit conversion between different solvers during the self-consistent iteration procedure.
Based on these considerations, we first examine the cancellation of self-interaction artificial
energy, then continue to check the data translation and unit conversion between the PB solver
and SIESTA. Finally, we demonstrate the overall accuracy of our model in the calculation
of solvation free energies, as well as the solvent effect on the solute electronic structure, by
a comparison with experimental data.

3.3.1

Validation of the cancellation of self-interaction energy

As described earlier, the use of the finite difference scheme in the solution of the PoissonBoltzmann equation results in the artifact of self-interaction energy which needs to be removed. Although it is common to assign the partial charge at the center of each atom
to mimic the effect of electrostatic interactions, this approach also has a similar issue of
artificially introduced self-interaction energy using the finite different scheme. It is known
that the cancellation of self-interaction energy works fine with the partial charge formalism
by computing the PB equation twice. However, the direct use of the quantum mechanical
charge density in the PB equation behaves differently. Because the magnitude of distributed
nucleus charges is much larger than that of partial charges, the direct use of the quantum
102

mechanic charge density leads to much larger self-interaction energy in the finite difference
scheme. Therefore, the accuracy of the PB solver for the reaction field potential and the
solvation calculation should heavily depend on the cancellation procedure. To our knowledge, with the direct use of the quantum charge density in the PB equation, no numerical
test has been done for the impact of self-interaction, neither has the performance of the
artifact energy cancellation been examined. To validate our approach, we test 16 small
molecules whose partial charges can be obtained from the literature. The details of structure data and parameter setting are described in Section 3.4. In particular, a uniform grid
size 0.25˚ is applied to the computation with the partial charge approach. However, in
A
quantum calculation the grid dimension is the same both in the PB solver and in SIESTA.
It is automatically generated in SIESTA by setting the MeshCutoff equal to 125 Rydberg.
Therefore, the grid size in the present simulation is no longer uniform and varies with different molecules. Take the water molecule as example, when cutoff energy is 125 Rydberg,
the grid size is hx × hy × hz = 0.1190˚×0.1192˚×0.1182˚, which is fine enough for the
A
A
A
solvation calculation. Table 3.1 and 3.2 list the total electrostatic energies both in vacuum
and in solution for these 16 molecules, together with the electrostatic solvation free energies
which are the difference between the total electrostatic energies in vacuum and in solution. It
is found that, as expected, the self-interaction energies with the quantum charge density are
much larger than those with the partial charge treatment. The former is hundreds of times
larger than the latter. However, through the cancellation, the electrostatic solvation free
energies are very close to each other. It can be concluded that most of the self-interaction
energy artifact can be removed in the present finite difference scheme with either the fixed
partial charge source or the charge density source. Therefore, the direct use of the quan-

103

tum charge density in the PB solver with the finite difference scheme is validated. Note
that although the total electrostatic energies vary under different mesh cutoff energies, the
resulting electrostatic solvation free energies show convergence.
Table 3.1: Comparison of total electrostatic energy (kcal/mol) and electrostatic solvation
energy (kcal/mol) obtained with the partial charge approach.
Compound
glycerol triacetate
benzyl chloride
m-bis(trifluoromethyl)benzene
N,N-dimethyl-p-methoxybenzamide
N,N-4-trimethylbenzamide
bis-2-chloroethyl ether
1,1-diacetoxyethane
1,1-diethoxyethane
1,4-dioxane
diethyl propanedioate
dimethoxymethane
ethylene glycol diacetate
1,2-diethoxyethane
diethyl sulfide
phenyl formate
imidazole

3.3.2

Vacuum
2443.64
210.88
1472.87
1068.47
866.22
315.5
1754.93
613.59
316.81
1726.22
517.65
1768.99
484.14
133.21
876.50
944.38

Solution
2456.10
215.93
1476.12
1077.68
873.91
319.77
1763.19
618.04
322.47
1734.06
522.18
1777.45
488.46
135.59
884.3
955.65

∆Gp
-12.46
-5.04
-3.25
-9.21
-7.69
-4.23
-8.25
-4.45
-5.65
-7.84
-4.55
-8.46
-4.32
-2.39
-7.85
-11.27

Validation of data translation and unit conversion

In this section, we demonstrate the reliability of data translation and unit conversion between
the PB solver and SIESTA by comparing the results from the current calculations with ones
from the literature. As stated before, the solution of the PB solver depends on the input
of the quantum charge density from SIESTA. The polarization of electron cloud, in turn,
requires the input of the reaction field potential obtained from the solution of the PB solver.
Therefore, the reliability of charge density data passed into the PB solver can be tested by
104

Table 3.2: Comparison of total electrostatic energy (kcal/mol) and electrostatic solvation
energy (kcal/mol) obtained with the direct use of charge density.
Compound
glycerol triacetate
benzyl chloride
m-bis(trifluoromethyl)benzene
N,N-dimethyl-p-methoxybenzamide
N,N-4-trimethylbenzamide
bis-2-chloroethyl ether
1,1-diacetoxyethane
1,1-diethoxyethane
1,4-dioxane
diethyl propanedioate
dimethoxymethane
ethylene glycol diacetate
1,2-diethoxyethane
diethyl sulfide
phenyl formate
imidazole

Vacuum
704426.41
341037.68
669772.11
533551.77
396325.28
478232.71
471728.52
332415.49
286515.14
434796.12
258225.34
434943.17
343486.83
223784.37
328961.95
199535.85

Solution
∆Gp
704439.14 -12.73
341043.97 -6.29
669774.74 -2.63
533563.01 -11.24
396335.07 -9.79
478239.05 -6.34
471736.99 -8.47
332421.71 -6.22
286521.15 -6.00
434805.20 -9.08
258230.45 -5.11
434952.17 -9.00
343493.67 -6.85
223788.70 -4.32
328968.86 -6.91
199549.02 -13.16

the solution of the PB equation. Meanwhile, the validity of reaction field potential values is
illustrated via the change of the solute self-energy due to the polarization of electrons. For
the purpose of a comparison, results by Wang et al [238] are used because of similar energy
decomposition in the solvation analysis. In their work, they studied the polarization of
electron cloud during the solvation process as well. A quantum mechanical calculation based
on density functional theory and the Poisson-Boltzmann equation was implemented. Finally,
by chemical analysis the solvation free energy was also decomposed into an electrostatic
solvation free term, a change term in the solute self-energy and a non-polar term. The
software UHBD together with a pre-determined solvent excluded surface was used for the
solution of the PB equation in their work [238]. Table 3.3 displays the comparison in solvation
free energy components for three small molecules which are water, NH3 and methanol. It is
105

evident that the results of solvation components from two different methods are comparable
to each other. This consistency proves the appropriate data translation process used in
different forms of computation domains, as well as the correct unit conversion between the
PB solver and SIESTA.
Table 3.3: Comparison of solvation energy components between present results and those of
Wang et al [238] for three small molecules.
∆Gp (kcal/mol)
∆GQM (kcal/mol)
Gnp (kcal/mol)
Compound OSMQ Wang
OSMQ
Wang
OSMQ Wang
Water
-7.55
-7.36
0.86
1.01
0.39
0.59
Methanol
-6.18
-5.53
0.63
0.77
0.57
0.77
NH3
-5.12
-6.77
0.87
0.92
0.46
0.63

3.3.3

Accuracy of solvation free energies computed by the present
model

Besides the data translation and unit conversion, the overall accuracy of the present model
still needs to be further verified by comparing with experimental data. In particular, it is of
crucial importance to check the accuracy of the total solvation energy as well as the solvent
effect on the solute electronic structure. The contribution of the solvation free energy from
the polarization of electron cloud can be decomposed into two parts. As shown in Table 3.4,
prior to the polarization, the solvent interacts with the solute based on its vacuum electronic
distribution which gives rise to the electrostatic solvation energy ∆Gv using the vacuum
p
charge density. When the polarization takes place, the electron cloud is redistributed to
reach a more favorable interaction with the solvent. This generates a gain for the solvation
free energy ∆∆Gp = ∆Gp −∆Gv . However, the redistribution of the electron cloud leads to
p
the change of the interactions between electrons and nuclei and those between electrons. It
106

causes an unfavorable decrease in the solvation energy (∆GQM ). Therefore, the total energy
contribution of the polarization is calculated by the sum of ∆GQM and ∆∆Gp . Table 3.4
summarizes the numerical results of the total solvation free energy and the decomposition
for the above mentioned three small molecules. Here, ∆Gtotal = ∆Gp + Gnp + ∆GQM .
As demonstrated in the table, the total solvation energies compare well with experimental
data [237]. Moreover, it is expected from the classical linear response theory that the loss
from the distortion of electron cloud is equal to about half of the gain from the solutesolvent interaction energy [224]. It is evident that our results are quantitatively in accord
with the theory. The slightly higher ratio (about 0.6) may be caused by the change in the
exchange-correction energy term within the DFT calculations.
Table 3.4: Solvation free energy (kcal/mol) and its decomposition.
∆GQM
Compound ∆Gv ∆Gp ∆∆Gp ∆GQM | ∆∆G | ∆Gtotal Exptl [237]
p
p
Water
-6.25 -7.55 -1.30
0.86
0.66
-6.31
-6.30
Methanol -5.18 -6.18 -1.00
0.63
0.63
-4.98
-5.11
NH3
-3.86 -5.12 -1.26
0.87
0.69
-3.79
-4.29

3.4

Applications

We consider three types of applications in this section. First, we apply our new multiscale
model to a set of 24 small molecules. Then, a challenging set of 16 molecules is studied.
Finally, three larger molecules are taken for efficiency and robustness test. The Dirichlet
boundary condition is used for both the generalized Poisson-Boltzmann equation and the
generalized Laplace-Beltrami equation as in our previous calculations [259, 252, 93, 20].

107

Table 3.5: Comparison of solvation free energies (kcal/mol) obtained from the present model
and experimental data for 24 small molecules.
Compound
Ethanol
Propionamide
H2 O
Phenol
Methanethiol
Propionic acid
Acetamide
Acetonitrile
Ethanethiol
Aniline
Methanol
Acetic acid
1-methylcytosine
Pyridine
9-methyladenine
1-methyluracil
NH3
4-cresol
4-Methylimidazole
Methylethyl sulfide
n-Butylamine
3-Methylindole
Methylamine
Benzene

Na
9
12
3
13
6
11
9
10
9
14
6
8
16
11
18
15
4
16
12
12
16
19
7
12

Gnp
0.74
1.02
0.39
1.21
0.7
1.02
0.82
0.87
0.87
1.23
0.57
0.83
1.32
0.98
1.45
1.28
0.46
1.35
1.04
1.03
1.08
1.42
0.54
1.20

∆Gp ∆GQM
-6.33
0.61
-12.19
2.14
-7.55
0.86
-7.88
1.01
-4.29
0.50
-7.73
0.87
-14.23
2.65
-8.23
1.36
-4.53
0.47
-9.34
1.15
-6.18
0.63
-8.67
1.03
-23.9
5.96
-7.61
1.49
-18.5
3.17
-15.35
2.78
-5.12
0.87
-7.94
1.00
-15.88
3.84
-4.83
0.55
-7.19
0.81
-10.40
1.66
-7.70
1.07
-4.69
0.51

∆Gtotal
-4.98
-9.03
-6.31
-5.66
-3.08
-5.84
-10.76
-5.99
-3.19
-6.97
-4.98
-6.81
-16.62
-5.14
-13.88
-11.29
-3.79
-5.58
-11.00
-3.25
-5.29
-7.32
-6.09
-2.98

Exptl [237] Error
-5.01
0.03
-9.41
0.38
-6.30
-0.01
-6.60
0.94
-1.24
-1.84
-6.47
0.63
-9.71
-1.05
-3.9
-2.09
-1.3
-1.89
-5.49
-1.48
-5.11
0.13
-6.7
-0.11
-18.4
1.78
-4.7
-0.44
-13.6
-0.28
-14.0
2.72
-4.29
0.50
-6.13
0.55
-10.25
-0.76
-1.49
-1.76
-4.29
-1.00
-5.91
-1.41
-4.5
-1.59
-0.9
-2.08

RMS error
Average error

3.4.1

1.31
1.06

Solvation free energies of 24 small molecules

Encouraged by the successful reproduction of the solvation free energies for the above
three small molecules, we apply the present differential geometry based multiscale model
and algorithms to the solvation analysis of an extended set of 24 small neutral organic
molecules. In this application, all geometric structures are taken from the Pubchem database
108

(http://pubchem.ncbi.nlm.nih.gov). The required pseudopotential input files for SIESTA in
the psf format are conveniently produced through a pseudopotential generator web (www.
tddft.org/programs/octopus). Atomic radii for the LB equation are adopted from a new
parametrization of ZAP-9 used by Nicholls [160] and in Chapter 2. Specifically, the radii of
hydrogen, carbon, oxygen, nitrogen, chlorine, fluorine and sulfur are set to be 1.1˚, 1.87˚,
A
A
1.76˚, 1.40˚, 1.82˚, 2.4˚
A
A
A
A

and 2.15˚, respectively. Note that different surface definitions
A

in implicit solvent models should have their own optimal radii set [235]. In particular, it is
found numerically that a continuous dielectric definition based model is supposed to have
radii of slightly larger values than those of a sharp interface based model [46]. Therefore,
we multiply the radii from ZAP-9 by a factor. In practice, a factor of 1.1 is used for all
atomic radii in a molecule of more than 14 atoms. However, if a molecule has less than 15
atoms, the factor is given by a formula 1.02 + (Max(0, Na -5))*0.008, where Na represents
the total number of atoms. Numerically it turns out that the solvation energy predictions
are sensitive to the radii factor for small molecules with less than 15 atoms. In this case, the
fewer atoms, the smaller factor.

In our calculation, since the polarization is treated explicitly with the quantum mechanical calculation, we set the dielectric constant in the solute region as ǫm = 1, while ǫs = 80
for the solvent region. Other parameters are set in a similar way as in Chapter 2: We choose
ρ0 /γ = 2 and take into account the pressure by setting p/γ = 0.2. Note that in the numerical
simulation, all ratio parameters here are treated as dimensionless. For L-J parameters, σs is
chosen to be 0.65˚ and σi is the solute atomic radii [235]. Due to the continuum represenA
tation of solvent in our model, the 6-12 Lennard Jones potential formula (A.2) differs from
the standard version— the distance used in our formula is no longer the distance between
109

the centers of solute atoms and the centers of solvent atoms but the distance between a
specific position in the solvent region and the centers of solute atoms. Therefore, the setting
of well depth εi differs from the ones taken from AMBER or OPLS force fields. As we did in
σi +σs 12
σi +σs 6
Chapter 2, it is determined by an equality, that is, εi
−2
= Di
|r−ri |
|r−ri |
if r is on the vdW surface of the atom. Here the constant Di should have different values
for various types of atoms. For simplicity we use a uniform constant D = 1.0 to determine
the value of εi . The grid dimension is the same both in the PB solver and in SIESTA, and
it depends on the MeshCutoff energy value in SIESTA, which is 125 Rydberg in current
simulations. The time stepping of τ = h2 /4.5 is used, where hx is the grid spacing at the x
x
direction. Finally, γ = 0.0065 kcal/(mol˚2 ) obtained from the previous work in Chapter 2
A
is applied to compute the total non-polar solvation energy.
Table 3.5 summarizes the numerical results of the solvation free energies of 24 molecules.
The root mean square error (RMS) of 1.31kcal/mol is obtained, which indicates a very
good agreement between the present prediction and experimental data [237]. The agreement
can also be seen from Figure 3.2. Moreover, to examine consistency, the ratio of ∆∆Gp
and ∆GQM is computed and listed in Table 3.6. It is evident that the numerical ratio is
always about 0.6. Therefore, the reliability and consistency of the present model have been
illustrated.

3.4.2

Solvation free energies of 16 molecules

Application of the set of 24 small molecules has shown that the proposed solvation model
works well for the solvation free energy calculation of small molecules. One of the motivations
in developing the present optimized surface model with the quantum charge density is to
110

OSMQ (kcal/mol)

0
−5

−10
−15
−20
−20

−15

−10
−5
Exptl (kcal/mol)

0

Figure 3.2: Correlation between experimental data and the present optimized surface model
with quantum correction (OSMQ) in solvation free energies of 24 small molecules.

(a)

(b)

(c)

(d)

Figure 3.3: Illustration of surface electrostatic potentials of four small compounds at their
corresponding isosurfaces S = 0.50. (a) Glycerol triacetate; (b) 1,1-diethoxyethane; (c)
Bis-2-chloroethyl ether; (d) Dimethoxymethane.

111

Table 3.6: Solvation free energy (kcal/mol) decomposition for a set of 21 molecules.
∆GQM
Compound
∆Gv ∆Gp ∆∆Gp ∆GQM | ∆∆G |
p
p
Ethanol
-5.42 -6.33
-0.91
0.61
0.67
Propionamide
-8.84 -12.19 -3.35
2.14
0.64
Phenol
-6.30 -7.88
-1.58
1.01
0.64
Methanethiol
-3.44 -4.29
-0.85
0.50
0.59
Propionic acid
-6.31 -7.73
-1.42
0.87
0.62
Acetamide
-10.13 -14.23 -4.10
2.65
0.65
Acetonitrile
-6.04 -8.23
-2.19
1.36
0.62
Ethanethiol
-3.77 -4.53
-0.76
0.47
0.62
Aniline
-7.52 -9.34
-1.82
1.15
0.63
Acetic acid
-7.01 -8.67
-1.66
1.03
0.62
1-methylcytosine
-14.68 -23.90 -9.22
5.96
0.65
Pyridine
-5.3
-7.61
-2.31
1.49
0.64
9-methyladenine
-13.42 18.50
-5.08
3.17
0.62
1-methyluracil
-10.88 -15.35 -4.47
2.78
0.62
4-cresol
-6.40 -7.94
-1.54
1.00
0.65
4-Methylimidazole -10.03 -15.88 -5.85
3.84
0.66
Methylethyl sulfide -3.94 -4.83
-0.89
0.55
0.62
n-Butylamine
-5.96 -7.19
-1.23
0.81
0.66
3-Methylindole
-7.80 -10.40 -2.60
1.66
0.64
Methylamine
-6.10 -7.70
-1.60
1.07
0.67
Benzene
-3.72 -4.69
-0.97
0.52
0.53

deal with a relatively challenging set of compounds, which was studied by Nicholls et al.
[160] and in Chapter 2 where the PB theory and fixed partial charges were used. This set
is challenging to compute because of the existence of polyfunctional or interacting polar
groups, which lead to strong solvent-solute interactions. The challenge has been illustrated
numerically in the previous work [160, 46]. In particular, with the OpenEye-AM1-BCC v1
charge and corresponding optimized ZAP 9 radii, the root mean square error (RMS) obtained
by Nicholls et al. is 1.71± 0.05 kcal/mol via an explicit solvent model. The smallest RMS
error of their single - conformer Poisson-Boltzmann approach is 1.87±0.03 kcal/mol [160].
By using our previous optimized surface model (OSM) with OpenEye-AM1-BCC v1 charges,
112

a better performance in the solvation calculation could be attained. However, RMS error was
still 1.76 kcal/mol. Large errors from benzamides can not be avoided if both OpenEye-AM1BCC v1 charge and ZAP 9 radii are used in the PB approaches. Errors from the calculation
of benzamides were still between 3.5 and 4.0 kcal/mol.
Therefore, hoping possible improvement, we introduce quantum mechanical corrections
to take care of the charge density. As we did before, structure data of this set of 16 molecules
is taken from the paper of Nicholls et al. [160]. In particular, atomic coordinates are taken
from their supporting information, which have been already optimized by using Gaussian03
package in vacuum with B3LYP/6-31G**. The atomic radii are still based on their new
parametrization ZAP-9 and multiplied by a common factor 1.1. All other parameters needed
in current model are set in the same way as described for the above set of 24 molecules.
Note that in the previous papers [160, 46], the set contains 17 molecules. Here we remove a
compound (benzyl bromide) because it has atomic species Br for which we can not obtain
a proper pseudopotential file from the mentioned pseudopotential generator website. Since
errors from the calculation of benzyl bromide previously was about 1 kcal/mol which is much
lower than RMS, exclusion of benzyl bromide should make the RMS increase in previous
results.

The results are summarized in Table 3.7 which lists the values of solvation free energies
for different components and gives a comparison of total solvation free energies between
calculated and experimental values [160]. It shows that the RMS error of 16 molecules from
the present model is 1.50 kcal/mol, which is much better than that from the explicit method
of 1.71 kcal/mol [160]. It is also better than our earlier OSM result of 1.76 kcal/mol in
113

Table 3.7: Comparison of free energies (kcal/mol) for 16 compounds.
Compound
glycerol triacetate
benzyl chloride
m-bis(trifluoromethyl)benzene
N,N-dimethyl-p-methoxybenzamide
N,N-4-trimethylbenzamide
bis-2-chloroethyl ether
1,1-diacetoxyethane
1,1-diethoxyethane
1,4-dioxane
diethyl propanedioate
dimethoxymethane
ethylene glycol diacetate
1,2-diethoxyethane
diethyl sulfide
phenyl formate
imidazole

Gnp
2.24
1.35
2.24
1.97
1.86
1.45
1.65
1.50
1.00
1.81
1.03
1.59
1.54
1.22
1.37
0.95

∆Gp ∆GQM
-12.73
1.35
-6.29
0.82
-2.63
0.36
-11.24
1.85
-9.79
1.52
-6.34
0.55
-8.47
0.92
-6.22
0.52
-6.00
0.66
-9.08
0.89
-5.11
0.48
-9.00
1.03
-6.85
0.61
-4.32
0.41
-6.91
0.89
-14.10
3.29

RMS error
Average error

∆Gtotal
-9.13
-4.11
-0.03
-7.42
-6.41
-4.34
-5.90
-4.20
-4.35
-6.38
-3.59
-6.38
-4.70
-2.69
-4.65
-9.86

Exptl
-8.84
-1.93
1.07
-11.01
-9.76
-4.23
-4.97
-3.28
-5.05
-6.00
-2.93
-6.34
-3.54
-1.43
-4.08
-9.81

1.50
1.08

Chapter 2. Therefore, a conclusion can be reached that nontrivial improvement is made
using the charge density directly computed from quantum mechanical calculations. Figure
3.3 depicts the surface electrostatic potentials of four compounds at their corresponding
isosurfaces S = 0.50. These surface potential profiles correlate with the surface electron
density distribution and chemical properties of the molecules.
Unfortunately, as shown in Table 3.7, errors from two amide compounds are still quite
large. This problem leads us to further explore the source of their errors. Note that in this
work, structural parameters are pre-determined and have not been optimized during the
quantum calculation of the electronic density profile. Therefore, the aforementioned large
errors must be due to the structural parameters. To prove this assumption, we carry out the
114

present solvation analysis of the 16 compounds based on the structural data provided from
the Pubchem database (http://pubchem.ncbi.nlm.nih.gov/). Our new results are listed in
Table 3.8. Indeed, errors from two amide compounds are significantly reduced. However,
the RMS error of the set (i.e., 1.50 kcal/mol) is exactly the same as that computed by using
ZAP-9 structural parameters [160], because larger errors from other compounds.
Note that our approach belongs to the so called “blind test” [160] in which the same set
of atomic parameters is used for all compounds. Certainly, our errors can be further reduced
if atomic parameters are chosen based on molecular information. As such, one is allowed
to use different atomic parameters based the chemical constitution and function groups of a
molecule. For example, carbons within the same molecule can have different atomic radii.
However, such approaches can no longer be called a “blind test” as discussed by Nicholls et
al. [160]

3.4.3

Solvation free energies of 3 larger molecules

The overall accuracy of the proposed model has been examined by the above two sets of
small molecules. Reliability, robustness and consistency are numerically illustrated as well.
As far as efficiency is concerned, the proposed model is expected to be slower than traditional
continuum models in the solvation analysis due to the involvement of QM calculations. However, the computational cost of the present model can be much less than the traditional QM
calculation and existing quantum mechanical continuum solvation models. This is due to the
following three reasons. First, implicit description of the solvent is adopted to dramatically
reduce the number of degrees of freedom. Second, the time-consuming quantum calculation
115

OSMQ/OSM/Nicholls’ (kcal/mol)

2
Exptl vs OSMQ
Exptl vs OSM
Exptl vs Nicholls

0
−2
−4
−6
−8

−10
−12
−12

−10

−8
−6
−4
Exptl(kcal/mol)

−2

0

2

Figure 3.4: Correlation between experimental data [160] and the present optimized surface
model with quantum mechanics (OSMQ) in solvation free energies of 16 compounds.

116

Table 3.8: Free energies (kcal/mol) for 16 compounds using structures from Pubchem data.
Compound

Gnp

∆Gp

∆GQM

∆Gtotal

Error

glycerol triacetate
benzyl chloride
m-bis(trifluoromethyl)benzene
N,N-dimethyl-p-methoxybenzamide
N,N-4-trimethylbenzamide
bis-2-chloroethyl ether
1,1-diacetoxyethane
1,1-diethoxyethane
1,4-dioxane
diethyl propanedioate
dimethoxymethane
ethylene glycol diacetate
1,2-diethoxyethane
diethyl sulfide
phenyl formate
imidazole

2.33
1.34
2.22
1.94
1.85
1.44
1.66
1.53
1.05
1.83
1.06
1.68
1.72
1.22
1.35
0.82

-13.01
-6.10
-3.29
-12.00
-10.59
-5.85
-8.51
-6.07
-5.13
-11.78
-4.61
-9.63
-5.68
-4.57
-6.61
-13.16

1.44
0.81
0.41
2.04
1.77
0.58
0.90
0.58
0.49
1.62
0.34
1.4
0.44
0.51
0.90
2.97

-9.23
-3.94
-0.66
-8.02
-6.97
-3.82
-5.94
-3.96
-3.59
-8.33
-3.21
-6.91
-3.52
-2.84
-4.36
-9.28

-0.39
-2.01
-1.73
2.99
2.79
0.41
-0.97
-0.68
1.46
-2.33
-0.28
-0.57
-0.02
-1.41
-0.28
-0.53

RMS error
Average error

1.50
1.18

has been accelerated by the pseudopotential and minimum basis set in the framework of a
linear scaling density functional theory in SIESTA. Furthermore, EPS charge fitting process, which depends on the definition of partial charges and the choices of sampling points,
is avoided in this model by the direct use of the charge density. Therefore, it is believed
that with powerful computer facilities, the current model can be a good choice to handle
complex systems such as large drug molecules, amino acids as well as moderately large
proteins. To examine the performance of this model for larger molecules, three molecules
are chosen, including phorbol (54 atoms), Staurosporine (66 atoms, a potent protein kinase
C inhibitor which enhances cAMP-mediated responses in human neuroblastoma cells), and
phorbol12,13-dibutyrate (71 atoms, an effective activator of calcium-activated, phospholipid117

(a)

(b)

(c)

Figure 3.5: Illustration of surface electrostatic potentials of three compounds at their corresponding isosurfaces S = 0.50. (a) Phorbol; (b) Phorbol12,13-dibutyrate; (c) Staurosporine.

dependent protein kinase C). Solvation free energies of these molecules are computed and
the computational cost is recorded with two quad-core Xeon 2.3 GHz processors. Our results
are listed in Table 3.9. It is also shown that the ratios of ∆∆Gp and ∆GQM are still about
0.6, which is consistent with those in smaller molecule calculations. In Figure 3.5, we illustrate the surface electrostatic potentials for the three compounds. The isosurface S = 0.50
is chosen for the plot. As shown in Chapter 2, different isosurfaces may exhibit different
electrostatic characteristic.

Table 3.9: Solvation free energies (kcal/mol) of 3 large molecules and corresponding CPU
time (hour).
∆GQM
Compound
∆∆Gp ∆GQM |
| Gnp ∆Gtotal Time
∆∆Gp
phorbol
-3.02
1.94
0.64
3.24
-15.53
0.44
phorbol12,13-dibutyrate -4.94
2.91
0.59
4.60
-17.43
0.97
Staurosporine
-8.39
4.78
0.57
3.79
-18.53
0.82

118

3.5

Chapter conclusions

The polar solvation energy was computed by the Poisson-Boltzmann equation with partial
charges adopted from molecular mechanical force fields. Generally, partial charges in force
fields are parameterized for certain class of molecules and may not be accurate for other
molecules. Additionally, the fixed charge pattern does not describe the charge arrangement
during the solvation process. This drawback limits the accuracy and utility of our earlier
solvation models. The present work addresses such a limitation of our differential geometry
based solvation models by the incorporation of quantum density into our earlier models.
We believe that solvation is subject to the fundamental law of physics. As such, all the
important state functions, including the quantum density of the solute, must be determined
by a multiscale total free energy functional, which has the contribution from the quantum
mechanical energy of electrons at the finest scale. Therefore, we construct a new multiscale
total free energy functional which includes the electron kinetic energies and potential energies. Apart from the earlier two governing equations, i.e., the generalized Poisson-Boltzmann
equation for the electrostatic potential and the generalized Laplace-Beltrami equation for the
solvent-solute boundary, one additional governing equation, the Kohn-Sham equation for
electronic structures, is also derived from the total energy functional by the Euler-Lagrange
variation. The solution of these three governing equations give rise to the desirable minimal
free energy of solvation.
Numerical methods and algorithms are discussed for the solution of three coupled partial
different equations (PDEs). The Poisson-Boltzmann and the Laplace-Beltrami equations
(i.e., a generalized Laplace-Beltrami equation) are solved in a similar manner as that in
Chapter 2. The Kohn-Sham equation is solved twice, once for the solute in vacuum and
119

the other in solution. By means of the reaction field potential, we can relate our KohnSham Hamiltonian in both situations to the standard one, so that existing computational
software packages can be utilized. The present work has developed a protocol to make use of
SIESTA (Spanish initiative for the electronic structure of thousands of atoms), an efficient
linear scaling DFT package, for the solution of the electronic density. Appropriate iteration
procedures are developed to dynamically couple three governing equations and ensure the
convergence of the solution.
The present multiscale model is validated by the solvation analysis of realistic molecules
whose experimental solvation free energies are available. We have particularly analyzed the
stability and consistency of the present model when atomic partial charges in our previous
Poisson solver are replaced by the continuous density distribution.

120

Chapter 4
Lagrangian formulation
This chapter presents a Lagrangian formulation of our differential geometry based solvation
models. Therefore, we consider the solvent-solute boundary as a 2-dimensional (2D) differentiable manifold embedded in a 3D Euclidean space or a hypersurface in a Riemannian
manifold. Then the subsequent free energy optimization can be carried out on the 2D manifold. For example, the area of a solvent-solute interface is modeled as a surface integration
over the biomolecular manifold. The Lagrangian representation of biomolecular surfaces has
a few utilities/advantages. First, it provides an essential basis for biomolecular visualization,
surface electrostatic potential map and visual perception of biomolecules. Additionally, it is
consistent with the conventional setting of implicit solvent theories and thus, many existing
theoretical algorithms and computational software packages can be directly employed. Finally, the Lagrangian representation does not need to resort to artificially enlarged van der
Waals radii as often required by the Eulerian representation in solvation analysis.
The main goal of the present chapter is to explore the Lagrangian formulation of the
differential geometry based solvation model, and analyze the similarity and difference of two
121

differential geometry based formulations. The mathematical analysis of biomolecular surfaces in the Lagrangian formulation is quite different from that of the Eulerian formulation.
The Lagrangian analysis of biomolecular surfaces makes the direct use of differential geometry theory of surfaces and manifolds. The surface minimization leads to the Laplace-Beltrami
operator, or the mean curvature operator. Whereas the Eulerian analysis of biomolecular
surfaces utilizes the coarea theorem of the geometric measure theory. The resulting operator
from surface area minimization can also be identified as a generalized Laplace Beltrami operator in a higher dimension. The connection of two representations is analyzed in this chapter.
The structure of governing equations, and the accuracy and efficiency of two formulations
are compared.

The rest of this chapter is organized as follows. In Section 4.1, we present the Lagrangian
formulation of differential geometry based solvation models. The total free energy functional
of solvation is constructed, and followed by detailed analysis of on-manifold variations. Such
variations produce coupled potential driven geometric flow and Poisson-Boltzmann equations. Section 4.2 is devoted to the computational methods and algorithms. We discuss
different realizations of biomolecular surfaces, including direct on-manifold evolution and
Eulerian embedding approaches. Algorithms for Eulerian and Lagrangian inter converting
are also discussed. A solution strategy for the coupled equations is designed and analyzed.
The proposed PDE methods are of second order in convergence. Method for solvation free
energy calculations is also given. The proposed methods and algorithms are validated in Section 4.3. A large number of numerical examples are designed to test the numerical accuracy,
convergence order and the computational efficiency of numerical methods and algorithms.
The proposed differential geometry based solvation model is applied to two classes of prob122

lems in Section 4.4. First, we examine the solvation free energy calculation of a set of
small molecules. We then extend our solvation calculations to a set of proteins. Results
are compared with experimental data, those obtained by using our earlier Eulerian formulation and those obtained by the classic molecular surface definition. Finally, we consider
two cases of salt-regulated protein-protein interactions. The protein binding affinities are
computed by using the proposed new models. The resulting binding affinities compare well
with experimental data in the literature.

4.1

Theory and model

This section presents the differential geometry based solvation model. We first discuss a
few free energy functionals of solvation. These functionals may appear exactly the same as
the solvation energy expressions in the classic theory of solvation. However, a fundamental
difference is that, in the present approach, the solvent-solute interface will be obtained by
the variational principle, rather than by a prefixed surface, such as the van der Waals surface
or the molecular surface. The governing equations for the solvation system, including the
Poisson-Boltzmann equation and the generalized Laplace Beltrami equation, will be derived
from the first variation.

4.1.1

Solvation free energy functionals

This subsection presents a few solvation models and establishes the notations for the further
development of differential geometry based solvation analysis. A polar solvation model is described before a nonpolar solvation model is given. The total solvation free energy functional
is designed as the combination of the polar and nonpolar components.
123

4.1.1.1

Polar solvation functional

The free energy functional of the electrostatic system was given exactly by Sharp and Honig
[199], and Gilson et al [97]. A sharp solvent-solute interface is assumed in their free energy
expression

Gp =



λm ρm φ − 1 ǫ|∇φ|2 − k T λs
B
2
Ω

Nc
i=1



n0 e−φQi /kB T − 1  dr
i

(4.1)

where the involved notations are defined as those in Eulerian formulation, while the permittivity ǫ(r) (also called dielectric coefficient) and the ionic function λm (r) and λs (r) are
defined as [7, 143, 93]
ǫ(r) = ǫm λm + ǫs λs

(4.2)

and

λm (r) =



 1 r ∈ Ωm

, λs (r) =


 0 r∈Ω
s



 0 r ∈ Ωm

(4.3)


 1 r∈Ω
s

where, the computational domain is divided into two subdomains, Ω = Ωm ∪ Ωs , with Ωm
and Ωs denoting the solute and solvent accessible regions, respectively. The domains Ωm and
Ωs are separated by an interface Γ. Here, ǫm = ǫ0 εm and ǫs = ǫ0 εs are the permittivities
of the macromolecule and the solvent, respectively, where ǫ0 is the permittivity of vacuum
and εα (α = m, s) are relative permittivities. We treat εα as constants. Note that in the
classic PB theory, these functions explicitly depend only on the radii of the solute [7, 93] —
they do not depend on Γ. However, in the present theory, these functions not only implicitly
depend on the radii of the solute, but also explicitly depend on the interface Γ, which in
124

turn, depends on the total energy functional.

4.1.1.2

Total free energy functional of solvation

The electrostatic free energy functional is complemented by nonpolar free energy functional
described in Section 1.1.4 to give the total free energy functional of solvation for biomolecules
at equilibrium

ǫm
Gtotal = γ · Area + p · Vol +
|∇φ|2 dr
ρs Uss dr +
ρm φ −
2
Ωs
Ωm


Nc
 ǫs |∇φ|2 + k T
−
n0 e−φQi /kB T − 1  dr,
B
i
2
Ωs
i=1

(4.4)

where we have used the coefficient definitions provided in Eqs. 4.2 and 4.3 to simplify the
integrals. This total free energy functional might be subject to a variety of corrections and
modifications.

4.1.2

Surface variation

The total free energy functional (4.4) is an important component of the present differential
geometry based solvation model. However, it does not provide a protocol for practical
solvation analysis. This subsection describes the variation principle which leads to desirable
governing equations for the surface formation and evolution, and for the evaluation of the
electrostatic potential.
In our previous minimal molecular surface (MMS) model [18, 20], the surface variation
was accomplished via the Euler-Lagrange equation in the Eulerian representation. In the
present Lagrangian representation, we perform the optimization of Gtotal on manifold Ξ
125

with respect to the solvent-solute interface Γ. In the spirit of differential geometry, the
interface can be represented as a closed surface in the 3D Euclidean space and denoted as
Γ(u1 , u2 ), which depends on the two real parameters u1 and u2 . The solute region, Ωm (Γ)
δ(·)
and the solvent region, Ωs (Γ) can be regarded as functions of Γ(u1 , u2 ). We use
to
δΓ
denote the first variation of (·) with respect to surface definition Γ,
δGtotal
δ
1
=
γ · Area + p · Vol +
ρs Uss dr +
ρm φdr −
ǫs |∇φ|2 dr
δΓ
δΓ
2 Ωs
Ωs
Ωm

Nc
1
−
ǫm |∇φ|2 dr −
kB T
n0 e−φQi /kB T − 1 dr .
i
2 Ωm
Ωs
i=1
(4.5)

We set

δGtotal
= 0 to construct the governing equation that describes the optimized solventδΓ

solute interface. To carry out this variation, we express the surface area and volume as the
following integrals
Area =

Ξ

dσ

and

Vol =

Ωm

dr,

(4.6)

where dσ represents the infinitesimally small surface element on the solute-solvent interface.
The complete first variation formula of G can be obtained by adding up the variation of
each term from Eq. (4.5). To this end, we consider a surface element f(u1 , u2 ) and its
infinitesimal displacement in the normal direction

f (ε) (u1 , u2 ) := f(u1 , u2 ) + εϕ(u1 , u2 ) · N(u1 , u2 )

(4.7)

where N is the outward unit normal direction and ϕ is an arbitrary C 2 function. In other
words, we consider a one-parameter family f (ε) of surface elements and the unperturbed

126

surface element is a particular case, f = f (0) . The tangent vectors of f (ε) are given by
∂f
∂ϕ
∂N
∂f (ε)
=
+ε
N + εϕ
.
∂ui
∂ui
∂ui
∂ui

(4.8)

To analyze the impact of the perturbation, we examine the first fundamental form of differentiable manifolds

(ε)
gij =

∂f (ε) ∂f (ε)
,
∂ui ∂uj

= gij + 2εϕ

(4.9)

∂f ∂N
,
∂ui ∂uj

+ ε 2 ϕ2

∂N ∂N
,
∂ui ∂uj

+

∂ϕ ∂ϕ
∂ui ∂uj

= gij − 2εϕhij + O(ε2 ).
We therefore have

(ε)
∂gij

= −2ϕhij .

∂ε

(4.10)

ε=0
By virtue of surface elements f (ε) and the first fundamental form, we can carry out the

127

surface variation of the area as follows

δ
δA
=
dσ
(4.11)
δΓ
δΓ
Ξ
δ
√
=
gdu1 du2
δΓ
U
∂
=
g (ε) du1 du2
∂ε ε=0 U
∂
g (ε) du1 du2
=
∂ε ε=0
U


(ε)
(ε)
(ε)
∂g
∂g
∂g
1
 du du
=
+ g11 22
− 2g12 12
√ g22 11
1 2
∂ε
∂ε
∂ε
U 2 g
ε=0
ε=0
ε=0
1
√
= −
ϕ (h11 g22 + h22 g11 − 2h12 g12 ) gdu1 du2
U g
√
= −
ϕ2H gdu1 du2
U
= −

Ξ

ϕ2Hdσ,

2
where g is the Gram determinant, g = Det(gij ) = g11 g22 − g12 and H the mean curvature
1
H = 2g (h11 g22 + h22 g11 − 2h12 g12 ), where gij and hij are defined in Appendix C.
The first variation of the volume enclosed by the manifold Ξ is derived as follows. We
first express the volume enclosed by the surface element f (ε) as a Taylor expansion in terms
of ε
Vol(f (ε) ) = Vol(f) + ε

δVol
+ O(ε2 ).
δΓ

(4.12)

As shown in the above calculation, we can pursue the volume variation with respect to Γ by
means of the variation with respect to ε
δVol
∂Vol(f (ε) )
=
δΓ
∂ε

=
ε=0

∂(Vol(f (ε) ) − Vol(f))
∂ε

128

.
ε=0

(4.13)

It follows [24]

δVol
=
δΓ
=
=
=
=

∂(Vol(f (ε) ) − Vol(f ))
∂ε
ε=0
√
ε
∂ 0 U DetJdu1 du2 dτ
∂ε
ε=0
√
DetJdu1 du2
U
τ =ε=0
√
ϕ gdu1 du2
U
Ξ

ϕdσ,

(4.14)

where matrix J is defined as






J =




∂f (τ ) , ∂f (τ )
∂u1 ∂u1
∂f (τ ) , ∂f (τ )
∂u2 ∂u1
∂f (τ ) , ∂f (τ )
∂τ
∂u1

and similarly, f (τ ) = f + τ ϕ · N,

∂f (τ ) , ∂f (τ )
∂u1 ∂u2
∂f (τ ) , ∂f (τ )
∂u2 ∂u2
∂f (τ ) , ∂f (τ )
∂τ
∂u2

∂f (τ ) , ∂f (τ )
∂u1
∂τ
∂f (τ ) , ∂f (τ )
∂u2
∂τ
(τ ) ∂f (τ )
∂f
,
∂τ
∂τ







,




(4.15)

τ ∈ (0, ε). We have carried out the calculation of inner

products
∂f (τ ) ∂f (τ )
,
∂ui
∂τ
where

∂f (τ ) , ϕN
∂ui

=
τ =0

∂f (τ )
, ϕN
∂ui

= 0,

(4.16)

τ =0

(τ )
vanishes since ∂f
and N are orthogonal to each other. Moreover,
∂ui

we have
∂f (τ ) ∂f (τ )
,
∂τ
∂τ

= ϕ2 ,

(4.17)

(τ )
where we use the fact that ∂f
= ϕN. Therefore, we can compute the determinant of the
∂τ

129

J matrix as





DetJ|τ =0 = Det 




∂f , ∂f
∂u1 ∂u1
∂f , ∂f
∂u2 ∂u1
0

∂f , ∂f
∂u1 ∂u2
∂f , ∂f
∂u2 ∂u2
0

0
0
ϕ2






 = ϕ2 g.




(4.18)

Moreover, the above derivation process can be extended to the first variation of a general
volume integral Ω F (r)dr.
m
δ

Ωm F dr
δΓ

=
=

√
ε
∂ 0 U F (u1 , u2 , τ ) DetJdu1 du2 dτ
∂ε
Ξ

(4.19)
ε=0

F (u1 , u2 )ϕdσ

where F represents a general integrable function of u1 and u2 , and is defined in the whole
domain. Similar to the volume variation, we end up with a surface integral.

Furthermore, because Ω F dr is independent of the surface variation, we have
δ( Ω F dr)
= 0.
δΓ

(4.20)

Then, it follows that
δ

δ( Ω F dr − Ω F dr)
Ωs F dr
m
=
=−
F ϕdσ.
δΓ
δΓ
Ξ

(4.21)

Therefore, the first variation of all other volume integration terms in Eq. (4.5) can be
130

attained by replacing F with appropriate integrands in Eq. (4.19) and Eq. (4.21)
δ

(4.23)

2
Ωm ǫm |∇φ| dr
=
ǫm |∇φ|2 ϕdσ,
δΓ
Ξ

δ

(4.22)

2
Ωs ǫs |∇φ| dr
=−
ǫs |∇φ|2 ϕdσ,
δΓ
Ξ

δ

Ωm ρm φdr
=
ρm φdσ,
δΓ
Ξ

(4.24)

Ωs ρs Uss dr
=−
ρs Uss ϕdσ
δΓ
Ξ

(4.25)

δ

and
δ

Ωs kB T
Nc

= −

Ξ

kB T
i=1

Nc n0 e−φQi /kB T − 1 dr
i=1 i
δΓ

(4.26)

n0 e−φQi /kB T − 1 ϕdσ,
i

where ∇φ on the surface represents the first derivative of potential φ.
Substituting Eqs. (4.11), (4.14), (4.22) (4.23), (4.24), (4.25) and (4.26) into Eq. (4.5)
yields
δGtotal
=
δΓ

1
1
−2γH + p − ρs Uss + ρm φ + ǫs |∇φ|2 − ǫm |∇φ|2
2
2
Ξ

Nc
+kB T
n0 e−φQi /kB T − 1  ϕdσ = 0.
i
i=1
131

(4.27)

Since ϕ is an arbitrary function, the following condition must be satisfied for each point on
the optimized interface

1
1
Wn ≡ −2γH + p − ρs Uss + ρm φ + ǫs |∇φ|2 − ǫm |∇φ|2
2
2
Nc
+kB T
n0 e−φQi /kB T − 1
i
i=1
= 0

(4.28)

It is noted that this condition was also obtained in our earlier work, but was derived in the
Eulerian representation using different mathematical techniques [243, 46].

4.1.3

Governing equations

The directional derivative of solvation free energy functional G in the direction of a normal
variation ϕ can be expressed as

Dϕ Gtotal (f) =

∂Gtotal (f (ε) )
∂ε

=
ε=0

Ξ

Wn ϕdσ.

(4.29)

If we choose ϕ(u1 , u2 ) = −Wn , then
Dϕ Gtotal (f) = −

Ξ

2
Wn dσ ≤ 0

. This means that the total free energy decreases along the normal direction when ϕ(u1 , u2 ) =
−Wn until it reaches a local minimal. Therefore the evolution f (ε) = f − εWn N leads to a
steady state and associated solvent-solute interface with strictly smaller energy. This analysis motivates the following potential driven geometric flow equation for the optimization of
132

the solute-solvent interface
∂X
= −Wn N,
∂t

(4.30)

where X ∈ Ξ ⊂ R3 is a position vector on the evolving manifold Ξ. Equation (4.30) is a
Lagrange formulation of generalized geometric flows and its structure has been discussed in
our earlier work [17]. This approach is computationally efficient but may have difficulties in
handling topological changes during the biomolecule surface evolution. Numerical schemes
for the solution of geometric flow equations is similar to that in Appendix A.
The electrostatic potential φ is governed by the Poisson-Boltzmann equation for traditional continuum biomolecular electrostatics applications. In the present approach, the
Poisson-Boltzmann equation can be easily derived from the variation of the full free energy
functional in Eq. (4.4) with respect to the electrostatic potential φ for a fixed interface Γ via
the Euler Lagrange equation [199, 97, 243]

δGtotal
=0⇒
δφ

Nc
−∇ · (ǫ(r)∇φ) − λs (r)

i=1

Qi n0 e−φQi /kB T = λm ρm .
i

(4.31)

This Poisson-Boltzmann equation admits a sharp solvent-solute interface Γ and is quite
different from the generalized Poisson-Boltzmann equation derived in Chapter 2, which has
an optimized smooth surface (OSS).
Note that the Poisson-Boltzmann equation (4.31) and the potential driven geometric flow
equation (4.30) are fully coupled. The existence and the uniqueness of their solution under
the biomolecular context can be an interesting mathematical issue. In practice, these two
equations have to be solved in a self-consistent manner. This aspect is discussed in 4.2.2.
The solution of the Poisson-Boltzmann equation is subject to the far field boundary condi133

tion, which in practice can be approximated by the Dirichlet type of boundary conditions
commonly used in the continuum electrostatic modeling (see Eq. (B.5)).

4.2

Methods and algorithms

This section discusses the solution strategies for a pair of coupled equations, the potential
driven geometric flow and the Poisson-Boltzmann equation.

4.2.1

Interconversion between the Lagrangian and Eulerian representations

The generalized geometric flow equation (4.30) is in the Lagrangian representation which is
well suited for boundary element or finite element type of methods. Although this Lagrangian
formulation of geometric flow models is relatively easy to implement in many applications,
such as surface smoothing, it can lead to computational difficulties in the case where there
are topological changes, such as surface breaking or merging. These difficulties origin from
the fact of singularity development on the manifold which is supposed to be smooth and
differentiable. These topological changes commonly occur in biomolecular surface constructions and molecular dynamics applications. One way to overcome these obstacles is to use
the Eulerian formulation [245, 18, 19, 20, 17]. The essential idea of resolving the difficulty
of a “singular manifold” or non-smooth surface is to embed the problem in a higher dimensional space such that the embedded function is smooth and differentiable. Another way to
avoid the difficulty of singular manifolds is to use a hybrid Eulerian-Lagrangian approach
for the biomolecular surface generation. In such a hybrid approach, the surface evolution
is carried out mostly in the Lagrangian representation, but is temporally switched to the
134

Eulerian representation whenever there is a singularity development on the manifold. In the
rest of this subsection, we discuss computational tools for the interconversion between the
Lagrangian and Eulerian representations.

4.2.1.1

Embedding the Lagrangian dynamics into the Eulerian representation

To embed a Lagrangian operator into its Eulerian representation, we introduce arbitrary
hypersurface function S(r) with r ∈ R3 . The earlier function χ(u1 , u2 ) can be obtained by
solving S = 0. For example, if S = X0 x2 + Y0 y 3 + Z0 z + D0 , where, X0 , Y0 and Z0 are
constants, and z0 = 0, then one can set
χ(t) = z = −x0 x2 − y0 y 3 − d0 ,

Y
D
X
x0 = 0 , y0 = 0 , d0 = 0 .
Z0
Z0
Z0

(4.32)

Via this example, it is easy to verify that the unit norm vector defined in Eq. (C.5) can also
be expressed in term of S
N=

(−χ1 , −χ2 , 1)
∇S
=
.
√
g
∇S

(4.33)

Then the desired surface can be represented as a set of points with a constant value of
function S
Ξ = {r|S(r) = L}

(4.34)

where L is an isosurface value. By the chain rule

∂S
∂t

∂S ∂X
·
∂X ∂t
∂X
= ∇S ·
∂t
=

= −Wn ∇S · N,
135

(4.35)

where X is a 3D position vector confined to the manifold Ξ. Due to Eq. (4.33), one has

∂S
∂t

= − ∇S Wn
=

(4.36)

1
2γH − p + ρs Uss − ρm φ − ǫs |∇φ|2
2

Nc
1
+ ǫm |∇φ|2 − kB T
n0 e−φQi /kB T − 1 
i
2
i=1
∇S

where all terms should be expressed in terms of level surface function S. In particular,
the surface mean curvature H must be rewritten in terms of S. The explicit form of mean
curvature (C.7) implies the equality 2H = ∇ · N which gives
H=

1 ∂
2 ∂u1

χ
√1
g

+

∂
∂u2

χ
√2
g

1
= ∇·
2

∇S
∇S

.

(4.37)

One can easily verify this relation by Eq. (4.32). Equation (4.37) connects the Lagrangian
representation of the Laplace-Beltrami operator with its Eulerian representation. Eventually, the potential driven geometric flow equation in the Eulerian form is obtained for the
optimized solvent-solute interface

∂S
=
∂t
+
=

1
− p + ρs Uss − ρm φ − ǫs |∇φ|2
2

Nc
1
ǫm |∇φ|2 − kB T
n0 e−φQi /kB T − 1 
i
2
i=1
∇S
∇S γ∇ ·
+V
∇S
∇S

∇·

γ∇S
∇S

136

(4.38)

where

1
1
V = −p + ρs Uss − ρm φ − ǫs |∇φ|2 + ǫm |∇φ|2 − kB T
2
2

Nc
i=1

n0 e−φQi /kB T − 1 . (4.39)
i

Equation (4.38) is the same as the generalized geometric flow equation derived in Chapter
2 using the characteristic function and geometric measure theory . This consistency lays
the foundation for switches forwards and backwards between the Eulerian and Lagrangian
representations and the development of hybrid methods for biomolecular surfaces. In fact,
Eq. (4.38) has the same structure as that of our potential and curvature driven geometric
flows and V is essentially the generalized potential defined in our earlier work [17]. Eq.
(4.38) is subject to the similar boundary and initial conditions as those of geometric PDEs
described in our earlier work [18, 20, 17].

4.2.1.2

Transform from the Lagrangian representation to the Eulerian representation

Usually the Lagrangian representation of surfaces is expressed in the form of triangulations
[188]. To convert the Lagrangian representation of surfaces into the Eulerian representation,
specifically, a Cartesian grid, we need to register the set of intersecting points between the
surface and the Cartesian mesh. For the purpose of computing the electrostatic potential
from the Poisson-Boltzmann equation which admits discontinuous coefficients across the
solvent-solute interface, we have to calculate the norms at all of the intersecting points
as well. A numerical algorithm for this Lagrangian-Eulerian transformation of molecular
surfaces was developed by Zhou et al. [259], although the original paper did not provide
implementation details.
137

We first set up a plane equation for each triangle on the molecular surface. For each
plane equation, we compute all the intersection points between plane and the Cartesian
mesh. For this set of intersecting points, we further record the subset that are located
within the triangle. For this subset of intersecting points, we finally compute their norms as
well by the second order finite difference scheme [259]. This algorithm has been extensively
tested in our previous work [259, 252, 93, 20].

4.2.1.3

Transform from the Eulerian representation to the Lagrangian representation

Once the hypersurface function S(x, y, z, T ), where T is the stopping time, is obtained from
the potential driven geometric flow equation (4.38) the solvent-solute interface can be extracted easily as an isosurface, S(x, y, z, T ) = L, where isosurface value L = (1 − δ)S0 .
Here S0 is set as the initial amplitude in geometric flow equation and 1 > δ > 0. In our
earlier work [18, 20], we have chosen S0 = 1000 and a small δ. Recently, in our differential
geometry based multiscale models [243], we have designated S as characteristic function of
the solute and chosen 0 ≤ S ≤ 1. There is no need to specify L in such a formulation. In
the present work, we choose L = Smax /2, where Smax is the maximum of S. This choice is
computationally stable and delivers correct MMSs, when the potential term is absent. It is
to point out that S here is quite different from the S used in our Eulerian formulation [46].
In the present work, S only serves as a hypersurface function for evolving and extracting the
solvent-solute interface and can take any real value. Numerically, isosurface extraction can
be done by existing software such as MATLAB. However, for further electrostatic analysis,
we need a Cartesian representation of the interface locations and the associated norm values.
Therefore, we construct a stand-alone algorithm to extract interface information. To this
138

end, the marching cubes method is adopted [140]. For a given grid partition, the marching
cubes algorithm simply deals with a local meshing problem by processing each cell or cube
independently. Each vertex of a cube can be either greater or less than the threshold value
L, giving 256 different scenarios. In considering the symmetry and complementarity, there
are only 15 canonical configurations in each cell needed to be addressed for the local meshing
[140, 67] A look-up table is a quite efficient local triangulation or Cartesian algorithm. The
marching cubes method can be modified in many ways to improve the accuracy, efficiency,
robustness, and topological correctness. Auxiliary binary tree structures are typically employed in the range-space approaches, such as kd-tree method and interval tree method, to
speed up the marching cubes method. For a structured grid dataset, geometric searching
methods exploiting spatial coherence can be simpler and more efficient than the range space
approaches.
To implement this scheme, first all points inside or outside the surface must be identified
according to S value in the Cartesian grid domain. The surface must intersect those cube
edges where one vertex is outside and the other is inside the surface. Therefore the surface
intersection points and their normal directions can be approximated by linear interpolation.
For instant, to compute an intersection point ro between an inside grid point r1 with value
S1 and an outside grid point r2 with value S2 , the distance d between ro and r1 is calculated
by

L = S1 ∗ (1 − d) + S2 ∗ d
d =

(4.40)

S1 − L
S1 − S2

where L is the isosurface function value. Obviously, with known positions r1 and r2 , distance
139

d determines the position of the intersection point ro . To calculate the norm vector at ro ,
we need to compute the normal vectors at r1 and r2 . In general, the normal direction of a
grid point (xi , yj , zk ) can be estimated by
Si+1,j,k − Si−1,j,k
2∆x
Si,j+1,k − Si,j−1,k
ny (xi , yj , zk ) =
2∆y
Si,j,k+1 − Si,j,k−1
nz (xi , yj , zk ) =
2∆z

nx (xi , yj , zk ) =

(4.41)

n = (nx , ny , nz )

where nx , ny and nz are the x, y and z components of the norm, respectively. Thus, the
norm at the intersecting point ro , denoted as no , can be interpolated through n1 and n2 ,
the norms of r1 and r2 , respectively

no = (1 − d)n1 + dn2 .

(4.42)

Clearly, the choice of L = Smax /2 offers the best computational accuracy and stability.
The unit norm No at the intersecting point can be easily computed as No = no . This
no
algorithm is used in our calculation of unit norms at the intersecting points. Obviously,
higher-order algorithms can be easily constructed when they are needed.

4.2.1.4

Numerical surface integral and volume integral in the Eulerian representation

Very often, we need to accurately carry out surface integration and volume integration in the
Eulerian representation. The surface integral of a density function f can be approximated
140

by [207]

Ξ

f (x, y, z)dσ =

Ω

f (x, y, z)δ(d(x, y, z))dr ≈

˜
f (xi , yj , zk )δi,j,k h3

(4.43)

i,j,k

where (xi , yj , zk ) is the coordinate of grid point (i, j, k), d(x, y, z) is the distance of a point
(x, y, z) defined in Ω from the interface Γ, h is the uniform grid size, and f (x, y, z) is the
˜
surface density function defined on Γ. The delta function δi,j,k is given by
˜
˜
˜
˜
˜(+x) + δ (−x) + δ (+y) + δ (−y) + δ (+z) + δ (−z)
˜
˜
δi,j,k = δ
i,j,k
i,j,k
i,j,k
i,j,k
i,j,k
i,j,k

(4.44)

˜(±α) , (α = x, y, z) are discrete delta functions [207]. To carry out integration exactly
where δ
i,j,k
on the interface, we use the following discrete surface integration formula [94]

Ξ

f (x, y, z)dσ ≈

f (xo , yj , zk )
(i,j,k)∈I

|ny |
|nx |
|nz |
+ f (xi , yo , zk )
+ f (xi , yj , zo )
h
h
h

h3
(4.45)

where (xo , yj , zk ) is the intersecting point of the interface and the x meshline that passes
through (i, j, k), and nx is the x component of the normal vector at (xo , yj , zk ). Similar
relations exist between (xi , yo , zk ) and ny , and (xi , yj , zo ) and nz . Since Eq. (4.45) has
already taken into account the contribution from irregular grid points outside the interface,
the summation is restricted to I, the set of irregular grid points inside or on the interface
[94]. The derivation of Eq. (4.45) is lengthy and is omitted here but it can be seen elsewhere
[94]. The surface area can be calculated by setting f = 1 in Eq. (4.45). The error of the
surface integration depends on the grid resolution and was observed to be approximately
second-order convergence [94].

141

The volume integral of the density function f can be simply approximated by [94]

Ωm

f (x, y, z)dr ≈

f (xi , yj , zk )h3

(4.46)

(i,j,k)∈J

where the summation is over J, the set of points inside the interface.

4.2.2

Dynamic coupling of the Poisson-Boltzmann and geometric
flow equations

The solution of potential driven geometric flow equation, or the generalized Laplace-Beltrami
equation, is discussed in Appendix A, including many discretization schemes. In general,
electrostatic energy is much larger than the non-electrostatic part so that the accuracy
of electrostatic potential calculation based on the Poisson-Boltzmann (PB) equation plays a
critical role in controlling the accuracy of the total solvation free energy. Therefore, numerical
methods that are able to deliver highly accurate solution of the PB equation is desirable.
In this work we apply matched interface and boundary method (MIB) to impose interface
conditions for high accuracy. This challenge is addressed in Section 1.5.3.
As described earlier, optimized electrostatic potential φ is obtained by solving the PoissonBoltzmann equation (4.31) in which solvent-solute interface Γ is used to determine ǫ and λ
values. The interface Γ is generated by the solution of the potential driven geometric flow
equation (4.38) which in turn depends on the electrostatic potential. Therefore, the geometric flow equation and the PB equation need to be solved simultaneously in the present
differential geometry based solvation model. In practice, this coupled nonlinear system can
be solved by an iterative procedure: First solving the PB equation with a fixed interface Γ for
142

φ; Then obtaining the interface Γ from solving the potential driven geometry flow equation
with a fixed potential φ. A more detailed algorithm is
1. Start with an initial solvent-solute interface, such as a solvent accessible surface.
2. Solve the Poisson-Boltzmann equation (4.31) for the potential with the initial solventsolute interface.
3. Obtain new solvent-solute interface by solving the potential driven geometric flow
equation (4.38) with the updated potential.
4. Calculate the solvation free energy with the resulting φ and Γ.
5. Go back to Step 2 until it converges.
The initial solvent-solute interface can be set by the solvent accessible surface with a probe
radius of 1.4 ˚. Another way to define the initial geometry is to use the interface obtained
A
from solving the potential driven geometric flow equation (4.38) without the potential term
V . Both approaches lead to the same result. In this study, we take the latter. The iteration
will be stopped if the values of total solvation free energy converge to within a designated
small criteria value which is 0.01kcal/mol for small molecules and 0.1kcal/mol for proteins
in this chapter. Formulism of solvation free energy evaluation is similar to that in Chapter
2.

4.3

Validation

This section is devoted to the validation of the proposed differential geometry based solvation model and a number of computational algorithms used in the present work. The
143

overall accuracy of solvation free energy calculation depends on the reliability and accuracy
of the solution of the geometric flow equation and the PB solver, surface and volume integrations, and the interface extraction process. The explicit Euler algorithm guarantees the
reliability and convergence of the solution of the geometric flow equation. The finite central
different scheme is of second order convergence in space and first-order in time although it
is computationally expensive [17]. The MIBPB-III has been verified to be of second-order
in convergence even with molecular surface singularities of proteins [93]. Therefore, it will
be second order accurate for the present application. In fact, the biomolecular surfaces
generated with geometric flows are free of geometric singularities, which is computationally
easier.
We first examine the impact of the interaction potentials to the surface morphology, and
surface electrostatic potentials. A few small molecular systems and 23 protein molecules are
used in this examination. We then check the behavior of the surface area under different
potential interactions. In particular, we verify whether the proposed minimal molecular
surface (MMS) [20] provides the extreme surface area for various molecules and proteins.
Finally, we investigate the convergence of the proposed iterative procedure for solving the
coupled Poisson-Boltzmann and geometric flow equations.

4.3.1

Validation of the interface extraction scheme

The numerical algorithm for surface integrations has second-order convergence [207, 94].
However, the MIBPB-III here has been modified for our purpose to admit the present optimized molecular surface (OMS) as the solvent-solute interface definition. This implies that
the reliability of the present MIBPB-III solver depends on the interface extracted by the
144

marching cube algorithm. Moreover, the performance of our surface integration and volume integration algorithms is also determined by the resulting interface instead of some
pre-determined interface such as the molecular surface [185]. Therefore, it is worthwhile
to validate the interface extraction procedure and algorithm in terms of surface area (˚2 ),
A
surface enclosed volume (˚3 ) and electrostatic solvation free energy (kcal/mol). In general,
A
there is no analytical result for electrostatic energy except for the one-atom system due
to Kirkwood [119]. For the one-atom system without interaction potential, the resulting
solvent-solute interface from the geometry flow evolution is a sphere with the same radius
as the initial one, for which the PBE admits an analytical solution. The surface area and
volume can be calculated analytically. Therefore, we consider a dielectric sphere of radii
2˚
A

p
with a unit charge at the center. We set γ = 0.5, S0 = 1000 and L = 500. Table 4.1

lists the numerical results under different grid resolutions h, which are compared with the
exact solution. The convergence in space is observed and a satisfactory result is attained.
Table 4.1: Comparison of surface areas (˚2 ), volumes (˚3 ) and energies (kcal/mol) for two
A
A
small systems.

h
0.5
0.25
0.125
Referenced Value

One Atom
Area Volume Energy
48.86
34.00
-84.92
49.04
33.56
-82.92
50.09
33.52
-82.08
50.265 33.510 -81.98

Diatom
Area Volume
95.24
71.00
99.28
72.73
100.5
73.20
100.34 71.18

Energy
-238.03
-233.66
-232.37
−233.67

Another test is done with a diatom system. It has been illustrated previously that
molecular surface of a diatom system can be reproduced by our PDE based approach at an
appropriate constant potential value [17]. In particular, when the van der Waals (VDW) radii
p
of two atoms are 2˚, the generated surface with γ = −0.222 will be almost identical visually
A
145

to the molecular surface with probe radius rm = 1.4˚. In this setting, the corresponding
A
solvation free energy, surface area and volume calculated by our numerical procedure are
compared with those based on molecular surface. To calculate the electrostatic solvation
energy, a unit charge is set at the centers of two atoms (-2.3,0,0) and (2.3,0,0). The numerical
results are summarized in Table 4.1. For a comparison, the reference molecular surface area
and volume of this diatom system are obtained by using the MSMS program [188] with probe
radius 1.4 ˚
A

and density 100. The electrostatic solvation energy (-233.67 kcal/mol) based

on molecular surface is calculated by the original MIBPB-III [93]. A good agreement is also
observed from this test.

4.3.2

Effect of interaction potentials

In this section, we illustrate the impact of potentials to the generation of solvent-solute
interface, consequently to the solvation analysis. Since the effect of the pressure term has
already been shown in our previous study, we focus our attention on the study of potential
effects which include short-ranged repulsive interaction, long-ranged attractive dispersion
interaction and electrostatic potential effect. Here we consider 12-6 Lennard-Jones pair
potential to model ViLJ . For the purpose of demonstration, all the surface profiles here are
constructed by using 6-12 decomposition and based on the geometric flow Eq. (4.36) in
absence of the pressure term and the ionic effect

∂S
=
∂t





1 + ∇S 2 ∇ · 

∇S
1 + ∇S



 + ρs V LJ −
γ
2
146



1 ǫs
1 ǫm
|∇φ|2 −
|∇φ|2  .
2γ
2 γ

(4.47)

Figure 4.1: Illustration of surface morphology of a diatom system with radii 1.87˚ and
A
coordinates (x, y, z) = (−2.2, 0, 0), and (2.2, 0, 0) under different solute-solvent interactions.
Top Left: The MMS (no potential); Top Right: The surface obtained under a repulsive
rep,LJ
potential (Vi
); Bottom Left: The surface obtained under a full L-J potential (V LJ );
Bottom Right: The surface obtained under a full L-J potential and an electrostatic potential.
It can be seen that the repulsive potential produces a “fat” surface, while an attractive
potential or an electrostatic potential leads to a “slim” surface.
Without any potential term, this geometric flow equation leads to the minimal molecular
surface (MMS) [20]. The effects of those three potentials are demonstrated by a diatom
system, a four-atom system and finally a protein molecule which is also used to illustrate the
potential impacts on surface electrostatic potential analysis. In the present computation, we
have treated the solvent density ρs as homogeneous.

4.3.2.1

Surfaces of a diatom system

We first consider a diatom system with van der Waals radius 1.87˚
A
(x, y, z) = (−2.2, 0, 0) and (2.2, 0, 0). Mesh size h = 0.04˚
A

and coordinates

is used. The L-J parameters

are set as follows: σi is taken from atomic radius and σs is chosen to be 0.65˚. Well depth
A
ρs
ǫi = 0.035 kcal/mol and bulk density coefficient γ = 2. To account for electrostatic
¯
147

potential effect, a unit charge is set on the center of each atom and we choose ǫs = 80 ∗ ǫs
γ
m
and ǫγ = 80 ∗ ǫm . We use ǫm = 1 and ǫs = 80 for dielectric constants in solute and solvent,
respectively. Figure 4.1 illustrates the different potential effects on the surface morphology
for the diatom system. We systematically change the potential effects to generate different
surfaces. We begin with no potential effect, which leads to the minimal molecular surface
(Top Left of Figure 4.1 ), then add the repulsive part of the L-J potential (Top Right of
Figure 4.1), then add an attractive interaction (Bottom Left of Figure 4.1) and finally add
the electrostatic potential effect (Bottom Right of Figure 4.1). It can be seen that the
repulsive potential produces a “fat” surface, while an attractive potential or electrostatic
potential leads to a “slim” surface. In other words, with a purely repulsive interaction
turning on, there is less bulk area between or around two spherical solutes, while more bulk
area with attractive potential or electrostatic potential turning on. This result is consistent
with experimental observations [236]

4.3.2.2

Surfaces of a four-atom system

The effects of potentials on the surface generation are further demonstrated by a fouratom system in Figure 4.2 with van der Waals radius 1.87 ˚
A

and coordinates (x, y, z) =

(−3.40, 0, 0), (3.40, 0, 0), (0, −2.94, 0) and (0, 2.94, 0). The needed parameters in Eq. (4.47)
are set as the same as the above diatom system except for setting 1/2 charge at the center
of each atom. We also systematically change the potential effects by beginning with no
potential which leads to the MMS in Figure 4.2(Top Left), then add the repulsive part of
L-J potential (Top Right of Figure 4.2), then add the attractive part in Figure 4.2(Bottom
Left) and finally add the electrostatic potential effect in Figure 4.2(Bottom Right). The
impact of potentials is similar to that in the diatom system. It is found that the size of hole
148

˚
Figure 4.2: Illustration of surface morphology of a four-atom system with radii 1.87A and
coordinates (x, y, z) = (−3.40, 0, 0), (3.40, 0, 0), (0, −2.94, 0) and (0, 2.94, 0) under different
solute-solvent interactions. Top Right: The surface obtained under a repulsive potential
rep,LJ
(Vi
); Bottom Left: The surface obtained under the full L-J potential (V LJ ); Bottom Right: The surface obtained under the full L-J potential and an electrostatic potential.
Again, it can be seen that the repulsive potential producess a “fat” surface; while an attractive potential or an electrostatic potential leads to a “slim” surface.

149

Figure 4.3: The projection of electrostatic surface potentials of protein 451c onto different
surfaces obtained under various solvent-solute interactions. Top Left: Attractive surface;
Top Right: The MMS; Bottom Left: Repulsive surface; Bottom Right: Polar surface. It is
noted that the repulsive surface is a “fat” surface comparing to the MMS; while an attractive
surface or a polar surface is a “slim” surface.

in the four atoms changes dramatically when varying non-bonding potentials. This would
imply that the size of pocket or cavity inside a protein can be dramatically changed under
different electrostatic potentials and/or solute-solute interaction potentials. Therefore, it
may result in a significant difference in physical properties of biological systems, which can
dramatically influence the selectivity and gating mechanism of ion channels.
150

Table 4.2: Electrostatic solvation free energies (kcal/mol), surface areas (˚2 ) and volumes
A
˚3 ) of protein 451c with different solvent-solute interactions.
(A
Surface
MMS
Repulsive
Attractive
Polar

4.3.2.3

Energy
-724.3
-635.2
-897.9
-838.1

Area
3695.0
3805.6
3904.9
3702.9

Volume
12881.9
13458.3
11635.6
12595.7

Surfaces and electrostatic potentials of a protein

Having illustrated the effects of various potentials to surface generations for simple artificial
systems, we now consider their impacts to the surface morphology and the solvation analysis of proteins. For this purpose, we choose a protein called heme-binding protein, Fe(II)
cytochrome C551 from the organism Pseudomonas aeruginosa (PDB ID: 451c). For the
structure, extra water molecules that are attached to proteins are excluded and hydrogen
atoms are added to obtain a full all-atom model. Partial charges at atomic sites and atomic
van der Waals radii in angstroms are taken from the CHARMM22 force field [146]. To show
the potential effects, each time we keep one and only one potential term in Eq. (4.47) to
produce a new surface which is used in our PB solver to calculate the electrostatic potential.
Starting with the MMS, the surface is called a repulsive surface when only a repulsive potential term is added, an attractive surface when only an attractive interaction is added and
a polar surface if only the electrostatic potential effect is allowed. The needed parameters
for potential expressions are set in the same way as in the 17 compounds which is described
in Section 4.4.1.1. Surface electrostatic potentials are plotted on the corresponding surfaces
in Figure 4.3. Meanwhile, electrostatic solvation free energies, surface areas and volumes
are calculated and listed in Table 4.2. Potential effects similar to the surface generations of
151

the diatom and four-atom systems are observed. Moreover, it is interesting to note that the
minimal molecular surface (MMS) has the smallest surface area among them so that it indeed minimizes the surface free energy. This is consistent with the mathematical proof that
the mean curvature flow equation leads to the minimal surface area. Yet, the MMS does not
possess the minimal volume. Instead, the attractive solvent-solute interaction leads to the
minimal volume. The repulsive solvent-solute interaction gives rise to the largest volume.
These results might appear to be counterintuitive. However, one has to keep in mind that
proteins are partially charged molecules. The electrostatic free energy plays a dominant role
in the solvent-solute interactions. There is an obvious correlation between the solute volume
and the electrostatic free energy: The larger the solute volume is, the smaller the electrostatic free energy is in absolute value. Therefore, the repulsive potential interaction leads to
the smallest electrostatic solvation free energy in absolute value, which is an indication of
the hydrophobic nature of the repulsive potential interaction. As expected, the attractive
solvent-solute interaction leads to the largest electrostatic solvation free energy, reflecting
the hydrophilic nature of the attractive solvent-solute interaction. It is believed that the
results in Table 4.2 are very helpful to the understanding of the sophistication of solvation.

4.3.3

Isosurface function value and minimal surfaces

The minimal molecular surface (MMS) proposed in our earlier work [20] was based entirely
on the differential geometry theory of surfaces. Although the minimal surface theory is
mathematically rigorous, the resulting surface might not be exactly the one with the minimal
surface area, when the evolution of the Laplace-Beltrami operator is carried out in the
Eulerian representation. This problem is due to the surface extraction process. There are
152

Table 4.3: Surface areas (˚2 ) for different surface definitions
A
Area
PDB-ID No.of atoms OMS MMS
1ajj
519
2201.4 2046.7
1bbl
576
2657.6 2434.1
1bor
832
2946.9 2683.9
1bpi
898
3274.9 3017.4
1cbn
648
2401.4 2212.7
1fca
729
2728.7 2474.1
1frd
1478
4467.2 3994.2
1fxd
824
3037.3 2762.5
1hpt
858
3368.3 3013.8
1mbg
903
3163.2 2831.3
1neq
1187
4829.0 4295.5
1ptq
795
2959.4 2685.8
1r69
997
3124.8 2806.3
1sh1
702
2808.4 2515.4
1svr
1435
4796.4 4247.9
1uxc
809
2916.1 2630.6
1vii
596
2571.2 2269.3
2erl
573
2380.4 2162.9
2pde
667
2782.0 2527.9
451c
1216
4184.7 3688.5
1a2s
1272
4507.5 3968.7
1a63
2065
7184.8 6369.7
1a7m
2809
7939.4 6918.9

infinitely many ways to select an isosurface value. Our tests indicate that the MMS can be
S0
obtained if we choose L = 2 . Therefore, we set the isosurface function value to 500 in the
present solvation free energy calculations rather than a value very close to 1000 which was
used in our earlier paper [20]. The results in Table 4.2 were obtained in this manner. In this
subsection, we further illustrate that the MMS indeed gives rise to the smallest surface area
for a set of 23 proteins. Moreover, we also study the impact of the pressure to the surface
area for a couple of given proteins.
153

We consider a set of 23 proteins in the present study. The detail of preparation and
treatment of protein data is described in Section 4.4.1.2. Two types of surfaces are generated
in the present work. The first type is the MMS constructed by the mean curvature flow [20].
The second type of surfaces is called optimized molecular surface (OMS) generated by using
the present differential geometry based solvation model. Results are listed in Table 4.3. As
expected, the surface areas from the MMS model are always smaller that those from the OMS
model. Essentially, the optimization of total free energy differs much from the minimization
of the surface free energy.
The possession of the minimal surface area in the MMS can be further demonstrated as
follows: we consider situations where only the constant pressure (p) is added into the mean
curvature flow equation to cause a perturbation of the MMS. For our purpose, two arbitrarily
chosen protein systems (PDB-IDs: 1ajj and 451c) from the set of 23 proteins are explored.
In our simulation, we set p=-0.4,-0.3,-0.2,-0.1,0,0.2,0.3,0.4 and 0.5. The minimal molecular
surface is obtained when p = 0. Figure 4.4 illustrates the difference of surface areas (˚2 )
A
between various resulting surfaces generated under different p values and the MMS for these
two protein systems. It is clearly seen that a small perturbation around MMS leads to a
larger surface area comparing to that of the MMS. In other words, the MMS indeed has the
minimal surface area.

4.3.4

Convergence of surface area, volume and energy

In this section, we also illustrate numerically the convergence and decreasing pattern of total
solvation free energy during the time integration, which has been shown theoretically in the
process of the model derivation. To this end, a small compound named diethyl propanedioate
154

Difference of surface area

250
200
451c
1ajj

150
100
50
0
−50

−0.4

−0.2

0
0.2
Pressure

0.4

0.6

˚
Figure 4.4: Difference of surface areas ( A2 ) between MMS and various resulting surfaces
generated under different constant pressure effects.

6
4
2
0
Energy
J(Volume)
Area

−2
−4
−6
0

10

1

10

N

2

10

3

10

Figure 4.5: Decreasing of surface area (×102 ˚2 ), J(volume)=(volume-2) (×102 ˚3 ) and
A
A
total solvation free energy (kcal/mol) in diethyl propanedioate as the number of iterations
increases.

155

has been chosen from a set of 17 compounds described in Section 4.4.1.1. The time history
of the total solvation free energies along with the evolution of solvent-solute boundary is
recorded. To illustrate the convergence pattern of the solvation free energy, we compute the
total solvation free energies in the intermediate states during the time evolution. The results
are shown in Figure 4.5. Here T denotes the time span and N = T represents the number
τ
of computational steps in the generalized geometric flow solver. In order to put surface area,
enclosed volume and total solvation energy together in one evolution picture, we illustrate
J(volume), which is a linear function of volume and shares the same pattern with volume,
rather than volume. It is found that the total solvation free energy decreases with respect to
the time evolution, which is consistent with our theoretical formulation. It is observed that
the solution of our model converges to a steady state in terms of volume (˚3 ), area (˚2 )
A
A
and total solvation free energy (kcal/mol).

4.4

Application

In this section, we consider the application of the proposed differential geometry model to the
calculations of solvation free energies and salt effects on the protein-protein binding affinity.
Previously, we have developed an optimized smooth surface (OSS) model in Chapter 2[46]
via the Eulerian formulation of the differential geometry based solvation model. It has been
demonstrated that OSS model successfully reproduces not only the solvation free energy of
small molecules but also the electrostatic solvation free energies of proteins. Although the
present optimized molecular surface (OMS) model is derived by using the same framework
of free energy functional optimization, the solvent-solute interfaces are entirely different in
two models. It is important to verify whether their results are consistent with each other.
156

For a comparison, we choose the same set of 17 compounds used in Chapter 2. Thus the
results from the OSS model are taken directly from the earlier work. In addition, we also
choose a subset of 23 proteins from 30 proteins studied in Chapter 2. The protein-protein
binding affinity is investigated by using two protein systems.

4.4.1

Free energy calculations

4.4.1.1

Solvation energies of 17 compounds

This test set of 17 small compounds was studied by Nicholls et al. [160] using a number of
approaches, including quantum mechanical methods, the PB theory etc. It is considered as
a challenging test set for computational methods because of the existence of polyfunctional
or interacting polar groups, which leads to strong solvent-solute interactions. The nonpolar
solvent-solute interaction potential in the present model provides a potentially efficient means
to deal with strong solvent-solute interactions.
In our calculations, we set the initial amplitude S0 = 1000 and isosurface function value
S0
L = 2 . Other parameters are set in the same way as that in our previous work. Again,
here γ (kcal/(mol˚2 )) serves as a fitting parameter and will have different values for differA
ent expressions of the nonpolar potential. Details are listed in Table 4.4. Typically, only
attractive solvent-solute interactions contribute to the dispersion effects in the third term of
Eq. (1.2). Here, we have three choices for the dispersion effect: V att,WCA , V att,6/12 and
V LJ . It turns out that the use of full L-J potential expression can offer the smallest root
mean square (RMS) error for the set of 17 compounds. Therefore, it will be chosen from
now on for the further study in this chapter except specified.
Structure and charge information of compounds are adopted from those in Nicholls’s
157

Table 4.4: RMS error with different nonpolar potentials.

Potential
γ
RMS (kcal/mol)

V att,WCA
0.0077
1.77

V att,6/12
0.0094
1.83

V LJ
0.0074
1.75

paper [160] too, which can be obtained from the supplementary information of the paper.
The results of the present full L-J potential model are summarized in Table 4.5. It gives
a comparison between the predicted and experimental values of solvation free energies of
17 compounds. The RMS error of the present model is 1.75 kcal/mol. This RMS error is
competitive with the explicit solvent approach (1.71±0.05 kcal/mol) under the same charge
and structure parameters set [160]. Moreover, it is interesting to note that the RMS is
almost the same as the one obtained from our earlier optimized smooth surface (OSS) model
using similar γ value (i.e., 0.0065 kcal/(mol˚2 ) vs 0.0074 kcal/(mol˚2 )). This consistency
A
A
can also be seen through Figure 4.6 which shows that the results from the OSS and the
present OMS are linearly correlated. The correlation coefficient is 0.999. It may reveal at
least two facts. First, in the framework of free energy optimization, the calculated results
using the Lagrangian representation and the Eulerian representation should be similar to
each other. Additionally, a satisfactory nonpolar term and the enforcement of the potential
driven geometric flow really play a critical role in the analysis of solvation free energies.

4.4.1.2

A set of 23 proteins

The set of 17 compounds has already shown the present approach’s ability to predict the
total solvation free energy of small compounds. Tests on proteins are needed to demonstrate
the capacity on the large system of interest. Encouraged by the success in the application to
158

0

OSS(kcal/mol)

−2
−4
−6
−8

−10
−10

−8
−6
−4
OMS(kcal/mol)

−2

0

Figure 4.6: Correlation of solvation free energy between previous optimized smooth surface
(OSS) model and the present optimized molecular surface (OMS) model in the set of 17
compounds listed in Table 4.5.

159

Table 4.5: Predicted and experimental total solvation free energies for 17 small compounds.
Compound

Gnp

glycerol triacetate
benzyl bromide
benzyl chloride
m-bis(trifluoromethyl)benzene
N,N-dimethyl-p-methoxybenzamide
N,N-4-trimethylbenzamide
bis-2-chloroethyl ether
1,1-diacetoxyethane
1,1-diethoxyethane
1,4-dioxane
diethyl propanedioate
dimethoxymethane
ethylene glycol diacetate
1,2-diethoxyethane
diethyl sulfide
phenyl formate
imidazole

2.33
1.39
1.36
2.22
1.99
1.91
1.44
1.67
1.55
1.01
1.87
1.02
1.62
1.57
1.22
1.37
0.80

∆Gp

∆G
(kcal/mol)
-12.36
-10.03
-4.87
-3.47
-5.06
-3.70
-3.30
-1.07
-9.22
-7.22
-7.84
-5.93
-4.16
-2.71
-8.21
-6.53
-4.63
-3.08
-5.64
-4.62
-7.75
-5.88
-4.64
-3.62
-8.40
-6.78
-4.40
-2.83
-2.40
-1.17
-7.82
-6.45
-11.56
-10.76

Exptl

Error

-8.84
-2.38
-1.93
1.07
-11.01
-9.76
-4.23
-4.97
-3.28
-5.05
-6.00
-2.93
-6.34
-3.54
-1.43
-4.08
-9.81

-1.19
-1.09
-1.77
-2.14
3.79
3.83
1.52
-1.56
0.20
0.43
0.12
-0.69
0.44
0.71
0.26
-2.37
-0.95

small compounds, we further consider a set of realistic proteins and compare the results with
those from previous optimized smooth surface (OSS) model and MIBPB-III [93] with predetermined molecular surfaces (MSs), which is defined as the inner surface smoothly traced
by a probe sphere as it rolls over the atomic sphere [185, 57]. Twenty three proteins, a test
set used in previous studies [17, 46], are chosen for the present calculations. All structures
and partial charges are obtained in the same way as the 451c system which is described
before. Table 4.6 shows the results of the present model, and those of the OSS and the
MIBPB-III. Results from the OSS and the MIBPB-III have been proved to be very close to
each other and they are competitive to those from quantum mechanic approaches [46]. Like
in the set of 17 compounds, results from the OSS and the OMS also show quite consistency.
The correlation coefficient between them are 0.999. This can also be observed in the Figure
160

0
OSS/MIBPB/MMS(kcal/mol)

OMS vs OSS
−500
−1000

OMS vs MIBPB
OMS vs MMS

−1500
−2000
−2500
−3000
−3500
−3500 −3000 −2500 −2000 −1500 −1000 −500
OMS (kcal/mol)

0

Figure 4.7: Correlation of electrostatic solvation free energy between the present optimized
molecular surface (OMS) model, and previous models, such as the optimized smooth surface
(OSS), the MIBPB-III and the MMS for 23 proteins listed in Table 4.6.

4.7. Therefore, this observation further convinces us that in the framework of differential
geometry based free energy optimization, the OSS and the OMS can be alternative to each
other in the aspect of solvation analysis. Since both of them share similar energy functional
expressions and take into account the key feature of total energy optimization at equilibrium.
If we remove the external potential effects in the surface evolution which are derived from
the energy optimization, the present OMS model returns to our previous minimal molecular
surface (MMS) model and the calculated results of solvation energies deviate dramatically.
Table Figure 4.8 demonstrates the difference of electrostatic solvation free energy between
the OSS and MMS models, as well as the difference between the OSS and OMS models. This
once again indicates the importance of polar-nonpolar coupling and solute-solvent interaction
in implicit solvent modeling and solvation analysis.
161

Table 4.6: Comparison of electrostatic solvation free energies of 23 proteins.
△Gp (kcal/mol)
PDB-ID No. of atoms MIBPB-III
OSS
OMS
1ajj
519
-1137.2
-1178.5 -1122.3
1bbl
576
-986.8
-965.9 -972.0
1bor
832
-853.7
-853.7 -836.3
1bpi
898
-1301.9
-1281.2 -1295.1
1cbn
648
-303.8
-304.8 -291.0
1fca
729
-1200.1
-1200.6 -1184.1
1frd
1478
-2852.2
-2844.8 -2846.7
1fxd
824
-3299.8
-3291.9 -3306.1
1hpt
858
-811.6
-808.2 -815.6
1mbg
903
-1346.1
-1328.2 -1346.9
1neq
1187
-1730.1
-1713.9 -1742.9
1ptq
795
-873.1
-866.2 -872.9
1r69
997
-1089.5
-1072.7 -1082.7
1sh1
702
-753.3
-771.8 -753.9
1svr
1435
-1711.2
-1704.6 -1716.7
1uxc
809
-1138.7
-1125.7 -1147.9
1vii
596
-901.5
-892.0 -907.0
2erl
573
-948.8
-935.8 -944.4
2pde
667
-820.9
-843.0 -812.3
451c
1216
-1024.6
-1020.6 -1016.8
1a2s
1272
-1913.5
-1900.3 -1902.8
1a63
2065
-2373.5
-2380.5 -2382.6
1a7m
2809
-2155.5
-2179.8 -2152.6

4.4.2

MMS
-921.0
-792.3
-665.9
-1060.0
-181.0
-1040.0
-2499.5
-3087.1
-570.0
-1148.7
-1401.6
-660.2
-824.4
-532.1
-1321.3
-919.3
-724.2
-812.2
-591.3
-718.2
-1633.0
-1851.0
-1699.9

Salt effect on protein-protein binding energies

Finally, we consider the application of our differential geometry based solvation model to
the calculations of salt effect on the protein-protein binding. This is the first time that
our new model is applied to the study of the salt effect. The ion concentration plays an
important role in the stability and even reactivity of biomolecules. This application can be
further extended to the binding affinity analysis of ligands, peptide, proteins, nucleic acids,
and membrane proteins. To this end, the potential terms caused by mobile ions need to be
162

600
500

OSS−OMS
OMS−MMS

Kcal/mol

400
300
200
100
0
−100
0

5

10
15
Protein

20

25

Figure 4.8: Difference of electrostatic solvation free energies between the OMS model and
previous OSS and MMS models for 23 proteins listed in Table 4.6.

restored in our calculation.
The full Poisson-Boltzmann (4.31) is coupled to the geometric flow Eq. (4.38) to obtain
the solvation free energy for proteins in the salted solvent. The solution procedure for the
nonlinear PB equation was described in our work [40]. The coupling of the nonlinear PB
equation and potential driven geometric flow equation is discussed in Section 4.2.2. For
low salt concentration and weak electrostatic potential, the linearized Poisson-Boltzmann
equation discussed in Section 4.1.3 can be applied.
For the binding free energy, only the electrostatic component and particularly, its salt
dependence are studied. The total binding free energy which includes many other terms that
do not depend on the salt concentration, does not need to be calculated. Then the electrostatic component of the binding energy (∆Gp ) is found as the difference of the electrostatic
163

Figure 4.9: Protein-protein complexes. Left: Protein complex 1emv; Right: Protein complex
1beb.

free energies of the complex and those of the free molecules

∆Gp (I) = GAB (I) − GA (I) − GB (I),
p
p
p

(4.48)

where GAB (I), GA (I) and GB (I) are the electrostatic free energies of the complex AB,
p
p
p
and the monomers A and B, respectively, at a given ionic strength, I. The salt dependence
of the binding free energy ∆∆Gp (I) is thus the difference in the electrostatic components
of the binding energies, Eq. (4.48), attained at some salt concentration I and at zero salt
concentration

∆∆Gp (I) = ∆Gp (I) − ∆Gp (I = 0)
= {GAB (I) − GAB (I = 0)}
p
p
−{GA (I) − GA (I = 0)}
p
p
−{GB (I) − GB (I = 0)},
p
p
164

(4.49)

10

Binding Free Energy(Kcal/mol)

Binding Free Energy(Kcal/mol)

15

Calculated
CF line
Experimental
EF line

5

0
−3

−2

Ln(I)

−1

0

1

Calculated
CF line
Experimental
EF line

−5

−10

−15
−5

−4

−3

−2
Ln(I)

−1

0

Figure 4.10: The salt dependence of the binding affinities of two protein complexes. Left:
Protein lemv; Right: Protein 1beb. Here OMS data are computed by our optimized molecular surface (OMS) model. NLPB data are taken from Bertonati et al’s paper [26].
where each energy term at different ionic strengths can be calculated via Eq. (2.23). In
general, the nonlinear PB equation should be used for the evaluation of salt effects on the
protein-protein binding. However, as shown by Bertonati et al. [26], both the linearized
PB (LPB) and the nonlinear PB (NLPB) can be applied to calculate salt effects when the
ionic strength of the salt is weak and the net charges of the binding complex and individual
molecules are relatively small. The results obtained with the LPB were very close to those
obtained with the NLPB. This encourages us to use LPB in this section to reduce the
computational complexity.
To test the utility of our new model in the calculation of salt effects on the protein-protein
binding, a hetro-dimeric and a homo-dimeric complex are selected for our study. These cases
were considered by Bertonati et al [26]. In the experiment, NaCl is used for the salt with a pH
value of 3. As shown in Figure 4.9, each protein complex encompasses two separated pieces.
The structures and charges of them are attained in the same way as earlier 23 proteins,
so are the needed parameters in the potential driven geometric flow equation. The salt
165

Table 4.7: Comparison of binding affinities of two proteins complexes from current simulations and those from Bertonati et al’s paper.
PDB
code

Complex
charge

Surface
Charge of the
˚2 ) free monomers
Area (A

1emv

-3

1465

1beb

+26

1167

Bertonati [26]
LPB NLPB

OMS

B=+5; A=-8

1.29

1.31

2.40

A=B=+13

-2.48

-1.53

-2.02

dependence of the binding free energy from NLPB simulation by Bertonati et al. as well as
our OMS model is shown in Figure 4.10, where the binding free energy ∆∆Gp (I) is plotted
as a function of the logarithm of the salt strength I. Additionally, binding affinities are
summarized in Table 4.7, in which the first four columns describe the properties of proteins
and the last two columns are the binding affinities extracted from the slope of the lines in
Figure 4.10. Note that the calculation is performed by assuming that all Arg, Asp, Glu and
Lys residues are ionized in both free and bound states. It is seen that our model clearly
reproduces the experimental observation, i.e., for the hetero-dimeric complex, the binding
free energies increase with the increasing ionic strength; while for the homo-diemric complex,
the affinity is negative. Moreover, as shown in the table the quantities of the binding affinity
obtained from simulations with the present OMS model are in good agreement with those
obtained by LPB and NLPB methods in Bertonati et al’s paper in which NaCl is used for
salt with a pH value of 3. Note that in the case of Lactoglobulin dimer, the results obtained
with all acidic groups neutral are shown.
Note that proteins considered in the present work have fixed conformations before and
after their interaction. Ions are treated as structureless. Therefore, the salt effect studied in
the present work does not include the effect of ions on biomolecule conformation and complex
166

formation due to different ion species [80, 225]. It takes more sophisticated models to study
the impact of the different ion species to molecular reaction rates and conformational change.
Such an aspect is beyond the scope of the present work.

4.5

Chapter conclusions

The objective of the present work is to explore an alternative formulation, the Lagrangian
formulation of differential geometry based multiscale solvation models. The Lagrangian representation of biomolecular surfaces is suitable for the visualization, surface electrostatic
potential map and visual perception of biomolecules. It is can be directly employed in the
implicit solvent models and existing software packages. Finally, the Lagrangian representation has an advantage that it avoids artificially enlarging van der Waals radii as often
required by smooth surface models [235].
In the present approach, the discrete and continuum domains are separated by a sharp
solvent-solute interface, which naturally constitutes a smooth and differentiable manifold
enclosing the biomolecule of interest. The time evolution of the manifold is governed by the
potential driven geometric flow, a mathematical framework introduced in our previous work
[242, 243, 17]. The specific potential driven geometric flow equation used in the present
work is derived via the first variation of the total free energy functional of solvation in the
Lagrangian representation. Such a derivation differs much from our earlier derivation using
the Eulerian representation and geometric measure theory [243].
Although there are some similarities in expressions of coupled PB equation and geometric
flow equation between our previous optimized surface (OSS) model in Chapter 2 and the
present optimized molecular surface (OMS) model, there are important differences to be
167

spelled out. First of all, the solute-solvent interface definitions in two models are fundamentally different. In the OSS model, the solute and solvent region is described by a continuous
characteristic function denoted by 0 ≤ S ≤ 1. In contrast, in the present OMS model, solute
and solvent regions are strictly separated by a 2D differentiable manifold. The function S in
Eq. (4.38) only serves as a hypersurface function for the geometric surface evolution. This
difference has a dramatic computational implication. The generalized Poisson-Boltzmann
equation with an OSS is much easier to solve than the OMS is. However, a formal comparison of this computational aspect is beyond the scope of the present work. Moreover, in
the potential driven geometric flow equation (4.38), ∇φ+ = ∇φ− because of the discontinuity of ∇φ inside and outside the solute-solvent interface. However, ∇φ+ = ∇φ− in the
overlap region of the OSS model due to continuous dielectric definition. Further, dielectric
constant ǫ(x) in the PB equation is defined in a totally different way: ǫ(S) is a function of
S in the OSS model, and there exists a smooth transition region from the low dielectric to
the high dielectric. In contrast, ǫ(x) is piecewise constant in the present model. In other
words, here ǫ = ǫs in solvent and ǫ = ǫm in solute, respectively. Further, a generalized
Poisson-Boltzmann equation was derived in the OSS model. Whereas, we formally end up
with the classical Poisson-Boltzmann equation in the present theory, although it is coupled
to the potential driven geometric flow equation. Yet, the present OMS brings up a number
of mathematical issues, including the singularity formation on the manifold, and Eulerian
embedding of Lagrangian dynamics. Finally, there are many computational problems associated with the Lagrangian formulation of our differential geometry based solvation model too.
For instant, the current discontinuous definition of ǫ leads to dramatic accuracy reduction
in the standard numerical schemes for the elliptic equations with discontinuous coefficients

168

and singular sources [254, 253, 258, 261, 260]. To overcome this difficulty, we have incorporated the highly accurate MIB scheme into our PB solver [259, 252, 93]. In addition,
many other computational issues, such as hybrid Lagrangian and Eulerian dynamics, level
set methods, isosurface extraction, surface integration, and Dirichlet to Neumann mapping
[93], are relevant in the present Lagrangian representation.

169

Chapter 5
Thesis achievements and future work

5.1

Contributions

As described in Chapter 1, solvation is an elementary process in nature that has a great
impact on many sophisticated physical, chemical, and biological processes. Therefore, the
importance of the understanding of solvation cannot be overemphasized. The major contribution of this thesis is in the construction of a series of novel differential geometry based
multiscale solvation models for chemical and biomolecular systems. We have extended our
earlier variational formulation of the surface free energy to the analysis of total solvation free
energy via the differential geometric theory of surfaces. As a key ingredient of the present
framework, the total energy functional encompasses coupled polar and non-polar contributions with a self-consistent interface definition. The true physical boundary of a biomolecule
in a solvent, as a physical concept, is determined by the optimization of the total free energy
for the solvation equilibrium. As such, a natural description of the solvent-solute interface is
provided by the differential geometric theory of surfaces and implemented by the generalized
170

Laplace -Beltrami equation, rather than simply ad hoc divisions of the solute and solvent
regions. Moreover, rigorous mathematical derivations have been demonstrated to obtain the
coupled PDE system in the spirit of the variational principle. Additionally, efficient and
robust computational algorithms have been designed for the 3D simulation. Finally, solvation analysis of both small compounds and proteins are carried out to further display the
accuracy, stability, efficiency and robustness of the proposed new models and the associated
numerical approaches. Comparison is made with both experimental and theoretical results
in the literature.
Biologically, this thesis provides a self-consistent treatment of the dielectric boundary in
all energy terms in the implicit solvent model based on the Poisson-Boltzmann theory.the
non-polar free energy terms. It simultaneously optimizes the total solvation free energy. This
is considered as a novel and important advance in the continuum treatment of molecular free
energies. Moreover, we propose a new theory that allows one to predict the solvation free
energy of a biomolecule in a computationally inexpensive way. The relative simplicity of
the theory could make this a useful daily tool for researcher to describe general trends in
solvation behavior of different biomolecular systems. Furthermore, our method potentially
allows one to investigate the ionic effects on the thermodynamics of biomolecules.
In our Eulerian formulation, the surface, separating the low dielectric interior from the
high dielectric solvent, is for the first time treated through a 3-D function S. S function takes
a value of 1 or 0 in the solute and solvent regions, respectively, and smoothly varies between
these values in the interface region. Then the van der Waals interaction, mechanical work
and the surface free energy terms are introduced as functions of S. The total energy can then
be optimized by simultaneous solution of the generalized Poisson-Boltzmann (GPBE) equa171

tion and the generalized geometric flow equation (GGFE) through the first order variation
with respect to the potential and S function. As such, realistic solvent-solute boundaries are
constructed. By solving the coupled GPBE and the GGFE, we obtain the electrostatic potential, the solvent-solute boundary profile, and the smooth dielectric function, and thereby
improve the accuracy and stability of implicit solvation calculations. We also design efficient
second order numerical schemes for the solution of the GPBE and GGFE. The matrix, which
results from the discretization of the GPBE, is accelerated with appropriate preconditioners.
An alternative direct implicit (ADI) scheme is designed to improve the stability of solving
the GGFE. Two iterative approaches are constructed to solve the coupled system of nonlinear partial differential equations. Extensive numerical experiments are designed to validate
the present theoretical model, to test computational methods, and to optimize numerical
algorithms.

Generally, partial charges from the existing force fields are parameterized for certain class
of molecules and may not be accurate for other molecules. Additionally, the fixed charge pattern does not describe the charge rearrangement during the solvation process. This drawback
limits the accuracy and utility of our earlier solvation models. The quantum formulation
work in this thesis addresses such a limitation by the incorporation of quantum electron
density in our earlier models. To this end, we initially construct a new multiscale total free
energy functional, which includes the electron kinetic energies and potential energies. By
means of the reaction field potential, we can relate the full Kohn-Sham Hamiltonian to the
standard Hamiltonian, so that existing computational software packages can be utilized. We
have developed a protocol to make use of the SIESTA (Spanish initiative for the electronic
structure of thousands of atoms), an efficient linear scaling Density Functional Theory pack172

age, to obtain the solution of the electron density. Appropriate iteration procedures are
developed to dynamically couple three governing equations and ensure the convergence of
the solution.
In the Lagrangian formulation of our differential geometry based solvation model, the
solvent-solute interface is modeled as a 2D manifold embedded into a 3D space. It is
suitable for visualization, surface electrostatic potential mapping and visual perception of
biomolecules. The Lagrangian formulation is can be directly employed in the implicit solvent model based existing software packages. Moreover, the Lagrangian representation has
an advantage in that it avoids artificially enlarging van der Waals radii as is often required
by smooth surface models. In this thesis, we also analyze the connections, similarities and
differences between the Eulerian and Lagrangian formulations of the solvation models. Such
analysis is important to the understanding of our differential geometry based solvation models. Finally, besides the solvation free energies, our Lagrangian formulation model is utilized
to evaluate the protein-protein binding affinities.
Most of the materials of this thesis are adopted from the following publications:
• Zhan Chen and Guo-wei Wei, “Differential geometry based solvation model III: Quantum formulation.”, submitted to Journal of chemical physics
• Zhan Chen, Nathan A. Baker and Guo-Wei Wei, “Differential geometry based solvation model II: Lagrangian formulation”, Journal of Mathematical Biology, in press,
2011.
• Zhan Chen, Nathan A. Baker and Guo-Wei Wei, “Differential geometry based solvation model I: Eulerian formulation”, Journal of Computational Physics, 229, vol. 22,
pp.8231-8258, 2010
173

The following publications are also closely related to the present thesis:
• Duan Chen, Zhan Chen and G.W. Wei, ”Quantum dynamics in continuum for proton
channel transport II: Variational interface”. in press, 2011
• Guo-Wei Wei and Zhan Chen , “Multiscale models for nano-bio systems”, Proceedings
of CMBE: ”2nd international conference on computational and mathematical biomedical engineering, 19-22,2011
• Duan Chen, Zhan Chen, Changjun Chen, Weihua Geng and Guo-Wei Wei, “MIBPB:
A software package for electrostatic analysis”, Journal of Computational Chemistry,
32 vol.4,pp.756-770, 2011 , 2010
• P. W. Bates, Zhan Chen, Y.H. Sun, G.W. Wei and Shan Zhao, ”Geometric and
potential driving formation and evolution of biomolecular surfaces”, Journal of Mathematical Biology, 59, vol.2, pp.193-231, 2009

5.2

Future work

In our current research, implicit solvent models assume local and linear solvent responses.
Further, we concentrate on the solvation calculations in the differential geometry based models. As far as our future work is concerned, models, numerical simulations and mathematical
proofs can be improved and extended as follows:
1. A simple homogeneous solvent density is employed in the present work. In future work
we will consider the implementation of the solvent variation. In particular, such a variation
can be computed by the integral equation approaches of solutions. A combination of the
integral equation theories with the present differential geometry based models will lead to
174

better predictions. This is because the non-uniform density fluctuations of solvent around
the solute-solvent interface can be taken into account.
2. Another important extension of the present work is the implementation of the implicit solvation model based molecular dynamics (MD). Currently, the Poisson-Boltzmann
(PB) based molecular dynamics algorithms have not been commonly used in the practical
simulation of macromolecules. Major hurdles to this development include limits in accuracy,
stability, speed and reliability. The multiscale models proposed in this thesis have given rise
to a new promise for the development of the PB based molecular dynamics. The accuracy and
stability problems in the previous PB based MD methods will not appear in our new model.
This is because they are free of interface singularity. Moreover, in all differential geometry
base models, the force expressions differ much from those in the classic Poisson-Boltzmann
based MD algorithms.
3. The resulting nonlinear PDE systems in our models pose challenges to mathematicians.
Numerically, we have shown the existence and local uniqueness of the solutions, which lead
to the optimization of the solvation free energy. Moreover, the convergence of the iterative
algorithms has been displayed numerically. However, rigorous mathematical proofs have not
been studied yet. In future work, upon the promise of numerical performance, we can study
these properties using rigorous mathematical tools.
4. The understanding of solvation is an elementary prerequisite for the quantitative
description and analysis of a variety of biological, physical and chemical processes. At an
equilibrium state, many models are expected to return to the solvation model. Therefore, our
differential geometry based solvation models can provide a classic framework for testing the
validity of other proposed implicit solvent based biological models, such as ion channel trans175

port models, proton transport models, protein-protein binding simulations, etc. Moreover,
it can be incorporated into other related models for accurate simulations in the future.

176

APPENDICES

177

Appendix A

Solution of the generalized
Laplace-Beltrami equation and the
ADI scheme

The solution of the generalized Laplace-Beltrami equations (also called the geometric flow
equation) Eq.(2.13) or Eq.(3.17) is studied here. First of all, to solve the geometry flow equation, the expression of solvent-solute interaction potential Uss must be prescribed explicitly.
Although Uss includes many unspecified interactions, we consider the following form

Uss (r) =

Ui (r),
i

178

(A.1)

where Ui (r) is the potential due to the ith atom in the solute molecule. One possible choice
of Ui (r) is the following Lennard-Jones (L-J) 6-12 pair potential
LJ
Ui (r) = εi

σi + σs 12
σi + σs 6
−2
|r − ri |
|r − ri |

(A.2)

where εi is the well-depth parameter, and σi and σs are solute atomic and solvent radii,
respectively. Here r is the point of interest and ri is a position vector of an atom in the
solute molecule. The L-J potential can be divided into attractive term U att and repulsive
term U rep in different ways. It can be a “6-12” decomposition as follows:
σi + σs 6
|r − ri |
σi + σs 12
.
|r − ri |

att,6/12
Ui
(r) = −2εi
rep,6/12
Ui
(r) = εi

(A.3)

Alternatively, it can also be a “WCA” decomposition based on the original WCA model
[130]


 −ε (r)
0 < |r − ri | < σi + σs
i
att,WCA
Ui
(r) =
 LJ
 U (r)
|r − ri | ≥ σi + σs ,
i

 LJ
 U (r) + ε (r) 0 < |r − r | < σ + σs
i
i
i
rep,WCA
i
Ui
(r) =

 0
|r − ri | ≥ σi + σs .

(A.4)

(A.5)

As indicated in Chapter 2, the WCA attractive potential provides good results for solvation.
Therefore, all the calculations in the present study are carried out by using the WCA decomposition. Note that due to the fast decay of the potential, only those solute atoms which

179

locate near the solvent-solute boundary are needed to include in the evaluation of Uss .
Additionally, a necessary step in solving our generalized Laplace-Beltrami equations is to
determine all the physical parameters involved. Because of the choice of the polar and nonpolar separation and the continuum representation of solvent in our model, not all parameters
from the literature are suitable. In particular, surface tension γ serves as a fitting parameter
in our model due to the ambiguity of its specific value in atomic-scale models [130, 160, 235].
Therefore, we rewrite the generalized potential driven geometric flow equation as

∂S
∂t

= |∇S| ∇ · γ
= |∇S|γ ∇ ·

∇S
|∇S|
∇S
|∇S|

+V

(A.6)

+ Vγ

where Vγ = V . Therefore, in addition to the Lennard Jones parameters εi , σs and σi , other
¯
γ
parameters including p/γ, ρs /γ, ǫs /γ, and ǫm /γ need to be pre-determined in the solution
of the generalized potential driven geometry flow equation.

The discretization scheme used here for the solution of the generalized geometry flow
equation (A.6) is similar to what we designed previously [20, 17]. It can be rewritten in the
form
2
2
2
2
2
2
(Sx + Sy )Szz + (Sx + Sz )Syy + (Sy + Sz )Sxx
∂S
=
2
2
2
∂t′
Sx + Sy + Sz
2Sx Sy Sxy + 2Sx Sz Sxz + 2Sz Sy Syz
2
2
2
−
+ Sx + Sy + Sz V γ ,
2 + S2 + S2
Sx
y
z

(A.7)

where t′ = tγ. To obtain the discretized form, we introduce the following notations. We
consider a discrete time tn := nτ where n is a non-negative integer and τ is the time stepping
180

n
size. We denote Sijk to be the discretized form of S(xi , yj , zk , tn ). An explicit scheme of
the generalized geometry flow equation is given by

n
2
2
2 n
n
S n+1 − Sijk := [vx δx + vy δy + vz δz ]Sijk + τ fijk ,
ijk

(A.8)

where

n
fijk =

Sx Sy Sxy + Sx Sz Sxz + Sz Sy Syz
−2
+
2
2
2
Sx + Sy + Sz

vx = τ

n
2
2
(Sy + Sz )
,
2
2
2
Sx + Sy + Sz ijk

vy = τ

n
2
2
(Sx + Sz )
,
2
2
2
Sx + Sy + Sz ijk

vz = τ

n
2
2
2
Sx + Sy + Sz V

n
2
2
(Sx + Sy )
,
2
2
2
Sx + Sy + Sz ijk

where
2 n
n
n
n
δx Sijk = (S(i−1)jk − 2Sijk + S(i+1)jk )/h2
2 n
n
δy Sijk = (S n
− 2Sijk + S n
)/h2
i(j−1)k
i(j+1)k
2 n
n
δz Sijk = (S n
− 2Sijk + S n
)/h2
ij(k−1)
ij(k+1)
{Sx }n = (S n
− Sn
)/2h
ijk
(i+1)jk
(i−1)jk
n
n
{Sy }n = (Si(j+1)k − Si(j−1)k )/2h
ijk
{Sz }n = (S n
− Sn
)/2h
ijk
ij(k+1)
ij(k−1)
181

ijk

n
n
n
n
{Sxy }n = (S(i+1)(j+1)k + S(i−1)(j−1)k − S(i+1)(j−1)k − S(i−1)(j+1)k )/4h2
ijk
n
{Sxz }Sijk = (S n
+ Sn
− Sn
− Sn
)/4h2
(i+1)j(k+1)
(i−1)j(k−1)
(i+1)j(k−1)
(i−1)j(k+1)
and

{Syz }n = (S n
+ Sn
− Sn
− Sn
)/4h2 .
ijk
i(j+1)(k+1)
i(j−1)(k−1)
i(j+1)(k−1)
i(j−1)(k+1)

Equation (A.6) is solved with the Dirichlet boundary condition S(r, t) = 0,

∀r ∈ ∂Ω.

For the initial value of S, we consider

S(x, y, z, 0) =



 1, (x, y, z) ∈ Dsa

(A.9)


 0, otherwise

where we define the domain enclosed by the solvent accessible surface to be Dsa =

Na
i=1 {r :

|r − Ri| < ri + rp }, with ri and rp being atomic van der Waals radius and the probe radius,
respectively. Here, Ri is the atomic center position vector of the ith solute atom and Na
denotes the total number of atoms for a given macromolecule. To protect the van der Waals
surface and make the computation more efficient, we only update the values of S(x, y, z, t)
at the points in between the van der Waals surface and the solvent accessible surface; i.e.,
(x, y, z) ∈ Dsa /DvdW , where DvdW is the domain enclosed by van der Waals surface
Na
DvdW = i=1 {r : |r − Ri | < ri }. Numerically, to avoid possible zeros in the denominator
of Eq. (A.6) we add a very small number, such as 10−7 , into the denominator expression,
which does not affect the result at all.
For simplicity, the widely used explicit Euler scheme can be applied to the solution of
the generalized Laplace-Beltrami equation for the time integration. The Euler scheme can
182

be combined with the second order central difference scheme for the spatial discretization
[20]. Nevertheless, this algorithm is not very efficient because a very small time stepping
size is required to guarantee the stability of the time integration. Therefore, an alternative
direction implicit (ADI) scheme is desirable. The ADI scheme is second order in both spatial
and time discretizations. It builds in a fast O(N) Thomas algorithm to solve the tridiagonal
linear system and thus is very efficient. The ADI algorithm is unconditionally stable and
allows a much larger time stepping size than does the explicit Euler scheme. An splitting
algorithm based ADI scheme reported in our earlier work [17] was the fastest scheme among
the tested ones under typical accuracy requirement for the mean curvature flow. Considering
the similarity of the current differential operator and the mean curvature flow, we adopt the
ADI scheme to speed up our generalized geometric flow solver. To this end, we modify Eq.
(A.8) as

1−

vx 2 vy 2 vz 2 n+1
vx 2 vy 2 vz 2 n
n
δx − δy − δz S
= 1 + δx + δy + δz Sijk + τ f (Sijk ) (A.10)
ijk
2
2
2
2
2
2

It follows that

Ax
2
Ax
1+
2

1−
=

Ay
2
Ay
1+
2

1−

Az
S n+1
ijk
2
Ax Ay Az
Az
n
n
1+
−
Sijk + τ f (Sijk )
2
4

1−

(A.11)

where
2
2
2
Ax = vx δx , Ay = vy δy , Az = vz δz

(A.12)

2 2
2
Here vx , vy , vz , δx , δy and δz are defined in Section 2.2.1. The following multi-step imple-

183

mentation can be carried out.
Step 1:

Ax
1−
2

n+ 1
3 =
S
ijk

1+

Ax
n
n
+ Ay + Az Sijk + τ f (Sijk )
2

(A.13)

Step 2:

Ay
1−
2

2
n+ 3
n+ 1 Ay n
3−
S
S
=S
ijk
ijk
2 ijk

(A.14)

Az
2

2
n+ 3 Az n
S n+1 = S
−
S .
ijk
ijk
2 ijk

(A.15)

Step 3:

1−

184

Appendix B
PB equation in different forms
Assuming that there are only two mobile ion spices and all ions are univalent, we can treat
them as positive and negative ions with charge +ec and −ec , where ec is the electron charge.
Then the nonlinear Poisson-Boltzmann (NLPB) equation (1.1) becomes [93, 108]

−∇ · (ǫ(r)∇φ) + κ2 (r)

kB T
ec

sinh

ec φ
kB T

= λm ρm ,

(B.1)

where κ is the modified Debye-H¨ ckel screening function describing ion strength and is
u
determined by
κ2 =

2λs Na e2
cI ,
s
1000kB T

(B.2)

where Na the Avogadro’s number, and Is the ion strength in the unit of mole. Numerically,
when T = 298K, the value of κ2 can be obtained via κ2 = 0.675365 ˚−2 Is . Note that
A
Debye-H¨ ckel parameter κ can also be expressed as [97]
u
Nc
2 = λs
κ
n0 Q2 .
i i
kB T
i=1
185

(B.3)

Equations (B.2) and (B.3) are equivalent to each other via the following formula [108]
Nc
1000M
1
n0 Q2 =
Is =
,
Na
2e2 i=1 i i
c

(B.4)

where M is the bulk concentration of ions in the unit of mole per cubic centimeter

mol for
cm3

both positive and negative ionic charges. Equation (B.1) is subject to the far-field boundary condition φ(∞) = 0. However, the Dirichlet boundary condition is used in practical
computations

Nm
φ(r) =

φi =
i

i=1

qi
e−κ|r−ri |/ǫs
ǫs |r − ri |

∀r ∈ ∂Ω,

(B.5)

where φi is the exact solution of a single ion in a homogeneous media. The linear superposition in Eq. (B.5) is very accurate if the macromolecule domain Ωm is sufficiently away from
the boundary ∂Ω.
Let define a dimensionless potential u through u = ec φ/kB T , one yields another formulation of the nonlinear PB equation in terms of u [40]

ec
λm ρm .
−∇ · (ǫ(r)∇u) + κ2 (r) sinh(u) =
kB T

(B.6)

If the potential is very weak, i.e., u ≪ 1, one can numerically solve the following linearized
PB (LPB) equation
ec
−∇ · (ǫ(r)∇u) + κ2 (r)u(r) =
λm ρm .
kB T

(B.7)

Note that in the Poisson-Boltzmann theory, there are two unit conventions in the literature that differs by a factor of 4π. Specifically, the convention used by Sharp and Honig
[200], and in some of our earlier work [93, 40] has a factor of 4π in the Poisson-Boltzmann
186

equation. Whereas, the convention used by Gilson et al. [97] and in our recent work [243]
as well as the present derivation, the 4π factor does not appear. Therefore, care is needed
in the comparison of the electrostatic potentials computed by these two conventions.

187

Appendix C

Differential geometry theory
preliminary

The solvent-solute boundary can be considered as a 2-dimensional (2D) differentiable manifold embedded in a 3D Euclidean space or a hypersurface in a Riemannian manifold. For
example, the subsequent free energy optimization can be carried out on the 2D manifold. Consider a C 2 immersion f : U → Rn+1 , where U ⊂ Rn is an open set and U
is compact[248]. Here f(u) = (f1 (u), f2 (u), · · · , fn+1 (u)) is a hypersurface element (or
a position vector), and u = (u1 , u2 , · · · , un ) ∈ U. Tangent vectors (or directional vectors) of f are Xi = ∂f , i = 1, 2 · · · n. The Jacobi matrix of the mapping f is given by
∂ui
Df = (X1 , X2 , · · · , Xn ). The first fundamental form is a symmetric, positive definite metric tensor of f, given by I(Xi , Xj ) := (gij ) = (Df)T · (Df). Its matrix elements can
also be expressed as gij =

Xi , Xj , where ,

i, j = 1, 2, · · · , n.
188

is the Euclidean inner product in Rn ,

Let N(u) be the unit normal vector given by the Gauss map N : U → Rn+1 ,
N(u1 , u2 , · · · , un ) := X1 × X2 · · · × Xn / X1 × X2 · · · × Xn ∈ ⊥u f,

(C.1)

where “×′′ denotes the cross product. Here ⊥u f is the normal space of f at point X = f(u),
where the position vector X differs much from tangent vectors Xi . The normal vector N is
perpendicular to the tangent hyperplane Tu f at X. Note that Tu f ⊕ ⊥u f = Tf(u) Rn , the
tangent space at X. By means of the normal vector N and tangent vector Xi , the second
fundamental form is given by

II(Xi , Xj ) = (hij )i,j=1,2,···n =

−

∂N
,X
∂ui j

.

(C.2)

ij

1
The mean curvature can be calculated from H = n hij g ji, where we use the Einstein summation convention, and (g ij ) = (gij )−1 .

For n = 2, which fits into our purpose, let us choose f(u) = (u1 , u2 , χ), where χ(u1 , u2 )
is a function of interest. We have the first fundamental form:



2 χ χ
 1 + χ1 1 2 
(gij ) = 
,
2
χ1 χ2 1 + χ2

(C.3)

∂χ
, i = 1, 2. The inverse matrix of (gij ) is given by
where χi =
∂ui



2 −χ χ
1  1 + χ2
1 2 
(g ij ) = 
,
g
2
−χ1 χ2 1 + χ1
189

(C.4)

where g = Det(gij ) = 1 + χ2 + χ2 is the Gram determinant. The normal vector can be
1
2
computed from Eq. (C.1)
N=

(−χ1 , −χ2 , 1)
,
√
g

The second fundamental form is given by (hij ) =

1
√ χu u
g i j

(C.5)

, i.e., the Hessian matrix of

χ.
The explicit form for the mean curvature operator can be written as

1
(h g + h22 g11 − 2h12 g12 )
2g 11 22
χ
χ
1 ∂
∂
=
.
√1 +
√2
2 ∂u1
g
∂u2
g

H =

(C.6)
(C.7)

In Section 4.1.3, we show that the mean curvature operator can be expressed in a (3D)
formulation.

190

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